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Quantum Sensor Networks (Lecture 1) by Saikat Guha

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after lunch

trying to keep awake to continue to do

so because we have got a

sort of change of gears and something

quite exciting coming up uh with uh

from the University of Arizona Tucson

I was going to talk to us about Quantum

sensor networks and he is sort of the

key man behind building large Quantum

networks in the US

so let's listen to thank you very much

Daniel thanks kasturi for inviting me

here and all the other organizers so

I'll try my best to do my part of

keeping you all awake

um and uh what I have decided to do I

made these slides after I heard both uh

size lectures and Carl's lectures uh you

will hear a little bit of a repetition

in terms of the the fundamental tools

behind sensing and Metrology but I'll

take a bend on where we will be doing

some math and actually applying the

tools on some problems

um so my first lecture today will be

focused on

developing some of those tools and

applying it to some problems in

photonic sensing and then we will look

at what you can do when you have

multiple sensors that are working

together to accomplish a task together

and in my second lecture we will be

talking about about a very specific application

application

um of a Quantum estimation Theory tools

for determining fundamental limits of

passive Imaging you're looking at

resolving stars for example in astronomy

or trying to resolve emitters in a

fluorescence microscopy setting

my name is said I'm at the University of

Arizona's College of optical sciences

and our we are located in a city called

Tucson in Arizona a few let's see if

all right yeah it works

um so Tucson is a city about 100 miles

south of Phoenix very close to the

Mexico border

the College of optical Sciences at the

University of Arizona and it's another

College focused in Optics in photonics

in Rochester so these are the two big

centers uh institutions in the U.S where

pretty much every single area of Optics

and photonics is represented all the way

from say high power lasers to non-linear

Optics Quantum Optics both Einstein

condensates every area of Optics

at the University of Arizona we are

I'll say a few words about the center

for Quantum networks that Anil pointed

out but I will not be talking about

networking but that's a huge Focus right

now within our Quantum information

science and engineering effort so these

are the faculty members and we are a

growing team and this last year we

launched a master's program in Quantum

information science and engineering and

we expect to offer that online or at

least parts of that online at some point

and I'll keep the organizers posted if

there is interest in this community

so let me say tell a few words about the

center for Quantum networks and then

I'll dive into the topic for today on

Quantum Metrology and sensing

so this Center is funded um as part of

the national science foundation's

engineering Research Center Program

there's a program called the ERC which

started in the 1980s and the objective

of that program was to take a technology

area that is right at the point where a

10-year long highly highly

transdisciplinary push and take it to a

point where that technology could be

ready for for further acceleration by

the industry and and by the society

so this is the first Quantum program as

that was funded as part of the ERC this

is about three years ago and it's a full

stack Quantum networking effort with the

objective of delivering fault tolerant

entanglement among distant locations

while serving multiple applications

simultaneously so just like the internet

supports many applications like the Zoom

call happening right now and the VoIP

and lots of different things happening

emails simultaneously the quantum

network should be supporting

entanglement delivery for so long it's

like telescopes and record 91 session

and maybe a blind Quantum computation

session all of those simultaneously so

we have people working all the way from

Quantum memory physics to spin Photon

interfaces to have the memories transfer

their qubits into the photonic domain

and vice versa Quantum error correction

to distal entanglement across a single

hop uh and then doing Quantum logic at

the memory nodes using uh nuclear spin

assisted hyperfine Logic on color

center-based Quantum memory since many

of these terms may not mean a lot to you

but I'll just I'll talk about some of

these Technologies a little bit later

and then ultimately once you have long

distance entanglement we are looking at

developing routing algorithms scheduling

algorithms Network tomography algorithms

and these are people who are working on

computer network Theory so it's all the

way from physics to computer science so

if you're interested in learning more

about our work in Quantum networking

definitely feel free to reach out to me

but with that I will switch to our topic

of discussion today

so to me this whole field of quantum

enhanced or Quantum inspired or Quantum

assistive sensing boils down to the fact

that you know life matter are they're

fundamentally quantum mechanical objects

even classical light that we are seeing

around us there is a Quantum State

description of this slide

it's a Romain phase insensitive

multimode gaussian State I mean it's

thermal light it's a it's a fancy way of

saying thermal light but but once you

cast light in the quantum mechanical

language you certainly have access to

some very very powerful tools

that allow you to do Quantum treatment

of information embedded in that light uh

and help you understand what is the best

possible Precision with which you can

measure certain quantities

now what does best mean

so that that that word best uh has to be

qualified what is your measure for

precision are you trying to tell apart

between a few different states of light

are you trying to estimate a parameter

embedded in a in a pulse of light

um are you trying to track how a

parameter changes in time

um are you trying to do a hypothesis

test between one known thing that you

are trying to discriminate between

absolutely anything else

all of those things that I just told you

about and there are more in the context

of communications there are different

Quantum information Quantum estimation

theoretic tools that tell you if that's

the parameter you care about how should

you measure that information bearing

light in order to get the best possible

best possible position for that

definition of test okay so that's that's

about the whole field of quantum

information Theory apply to photons now

another thing that I think is very

important to understand and we have

heard this in all of the talks in the

last couple of days but I still wanted

to stress it there is no way to evade

measurement noise there's no matter how

you choose to measure your light

there will be a fundamental noise I'm

not talking about you know excess noise

but there is something called short

noise so there's a Quantum limited noise

even when you are measuring uh squeeze

sliders a minimum amount of noise that

that quantum mechanics says that you

have to add okay so all of this entire

field of quantum Metrology to me it is

the it is a whole field that that that

looks at how can you engineer that noise

to your advantage by perhaps doing

something to the optical domain

information before you subject to that

inevitable noise perhaps you would

position that information in that light

in a more favorable way to that

inevitable noise that is yet to come so

that is about uh that that's how you

design optimal receivers or optimal

measurement strategies

so these Quantum Metrology tools give us

the fundamental limits of precision but

as uh you know Carl also pointed out in

his talk that oftentimes even though

these Quantum Metrology tools give you a

hint at how you we would design a scheme

that will actually get those limits

that going from understanding the

fundamental limits to finding out that

that measurement strategy is often very

very difficult

um and that itself is is a science and

art on its own right and we will see

some examples of those today

okay so with that Prelude about My

Philosophy about the field let me tell

give you an outline of what I'm going to do

so you will hear me talk about a whole

bunch of bounds you heard from both PSY

and Carl about the crema Rao bound

we'll talk about other Bayesian bounds

and estimation Theory

helps from bound chair enough bound to

level bound and all of these different

bounds they talk about you know

understanding what is the best possible

Precision for a particular definition of

the word best you define what you what

you want what you care about so in

today's lecture I will start with some

some primer on estimation Theory I will

take a more practice Bend where we will

be actually doing some math and doing

some calculations I'll spend a little

bit of time talking about the quantum

description of light and then I will

launch into applications first the

applications will be on uh you know

individual sensors and then I will go

into network of sensors and depending

upon where we finish today with that one

and a half hour mark I will might not

cross the line right here either like

might spillover or leave something for

the next lecture so we'll see how far we go

okay so let's go ahead and get started

um so we I'll first start with

estimation Theory

um and uh I would I'll put some pictures

on on um of various hikes my group loves

to do in the Tucson area and

people who have visited my group they

know that we would

take them out on Hikes too so if you if

you wish to come to Tucson we would love

to take you out you know see these um

these cactuses these are called the

saguaro cactus if you haven't seen them

they are um

grow up to 20 feet or so high

and they live for more than 200 years in

fact people who are experts on them they

can look at the you know the uh the

different widths of these cactuses along

with along its height and they know

which year Tucson got how much rainfall

so they're absolutely gorgeous scenery

so let's talk about estimation Theory

classical estimation here in this very

very basic simple construct X is my

parameter of Interest

but what you actually see is Y which is

a noisy version of x

and you know P of Y given X

and you know POI gave an X perhaps

because you know where X comes from and

what measurement you have choose Chosen

and why is the output of your measurement

measurement

your job is to estimate X and let's take

the uh the what I call the fisherian

point of view where X is a fixed

parameter that is what it is but you

don't know what it is there is no prior

probability distribution of it's called

the frequentest approach now

now

if you really wanted to estimate X

what's the best why you hope for

y equal to X well if if it would be

wonderful if y was actually X and you

could measure it right but then Y is a

noisy version of X so what you the next

best thing you do is that uh you from

your probability distribution you pick

the value of x

for the Y that you just got you pick

that value of x for which this

probability distribution P of Y given X

is the highest seems like the most

obvious natural thing to do

and this estimator of X and I'm using

the the hat so I use the inverted hat so

it's the same thing

um this is called the maximum likelihood estimator

estimator

and it's the most common sense estimator

but it also happens to be the estimator

that minimizes the variance of that

estimator so gives you the best quality

estimate of that of that parameter

um so this maximum likelihood estimator

um if you were to write down the

variance of absolutely any estimator

that you could get here from y so

estimator is simply a function that

learns this Y into your belief and

estimate about X and you might have n

trials getting your same parameter X is

going through multiple instances of this

probabilistic channel to get you these

different outcomes y1 y through to y n

and and this Y in general could be a

vector and the estimate is just one

let's say in this case a scalar

parameter that you are estimating

so absolutely any estimator satisfies

this Fischer information driven this is

the criminal outbound inequality

um and uh there is a so you saw some

versions of this criminal bound where

there was nothing in the numerator there

was one in the numerator and I believe

that PSI U had covered this bias term

um this m of X is the estimate the the

expectation of the estimator if this m

of X happens to be equal to the value of

the parameter then you have an unbiased

estimator in which case m is derivative

M of M of X is one so you get a one n is

the number of Trials and J of X is the

is the official information now let's

scare this facial information for a

moment and you have seen again this

written in multiple different ways in

the last few days look at this quantity

over here inside the square bracket it's

the derivative of the log of the

probability distribution

now if if y so this this condition just

the this this log of P or Y given X that

um is uh this is called the the

derivative of that is called this four

function by statisticians

if you get a y okay and if you change

your X

what kind of Y is a good y to estimate X

a y that actually changes as you change

X right because then you will be able to

estimate X

so if you have a value of y's such that

the probability distribution P or Y

given X doesn't change too much it is

not giving us much information about X

so so that's why if this partial

derivative magnitude squared if this

thing is small then that value of y is

not a good value of y in terms of its

ability to give you a good estimate of x

so that's why it seems reasonable that

the expected value of that squared

quantity which quantifies the the

goodness of that particular y

to give you an estimate uh

expectation taken over the probability

distribution of Y given the value of x

should be a quantity that tells you on

an average

uh what is the amount of information

that you're going to get from one sample

of Y about X now obviously I just said

it when Carl proved it and so I also

showed a proof for the quantum case but

that's the intuitive understanding of

why Fisher information is uh does what

it does if it's large it means your

variance scales more rapidly as n increases

increases

another thing that's important here is

that the maximum likelihood estimator if

you were to plug in this estimator over

here as n increases to Infinity this

bound is saturated and I'll talk more

about that

okay so now there is another view of the

world that that's called Bayesian

estimation Theory where you have a prior

probability distribution available on x

maybe because you have a knowledge of

the problem itself maybe you have made

prior measurements that you have used to

convert that to a prior probability

distribution that you could leverage or

maybe it is just the setting of the

problem setting that has been solved in

the past that you have some reason to

believe that you know something about X

before even you start to measure it now

if you have a prior probability

distribution then the quantity that you

care about minimizing is called the mean

squared error so I should have just

written NSE mean squared error and this

first m stands for the minimum mean

squared error so what I'm writing here

is the This Is The Joint distribution of

P of X and Y so you have P or Y given X

already given P of X is also given this

is your prior and this quantity is your

squared error and you are just taking an

average over the joint distribution so

that's called the mean squared error so

your job is to find the estimator for

for X that minimizes that mean square

error so what is that estimator that

estimator is given by the expected value

of x given y so what you do to calculate

this estimator so if I give you a p of I

given X and I give you a p of X what you

do is that you apply Bayes rule to take

these two and get P of x given Y and I

believe that you all here know the base

rule so you can go from P or Y given X

to P of x given y

and then if you think about this P of x

given y this is the probability that the

parameters value is x given that you

have observed y

so it's actually probably the

description of X like given y so if I

had if I integrate that x times PX it's

the average value of x right so this

thing is the average value of x

conditioned on the measurement outcome

that you have gotten

and that is the mmsc estimator so you

can prove that if you use this as your

estimator and plug it in here the value

of this mean squared error you would get

is the minimum possible value you could

have gotten

now even for classical estimation Theory

this thing is actually not easy to

calculate for all values of P of I given

X this estimator can also get very

complicated to calculate so people came

up with bounds to this and there's

there's many many bounds to the this

mmsc there's the van trees bound there's

the zip zakai bound there's there's many

different bounds this one that you

learned in textbook estimation here it's

called the Bayesian Cramer row bound

which is kind of a weird phrase I mean I

don't like that at all it's like

criminal bound is a fisherian bound

which you're saying sebation Kramer

outbound but that's what it is called

it's called bcrb and and why is it

called that if you look at this JB the

expression for that it looks identical

to the fission information except that

you replace the conditional distribution

by the join distribution everywhere this

one by the joint distribution this one

by the joint distribution and you can

prove that one over the JB

is a lower bound

to to the minimum to the mean squared

error and the thing is that this bcrb is

not saturable always meaning there are

problems I'll show for example for which

you calculate this bound and it's not tight

tight

okay so this is all I'll talk about

classical estimation Theory now this um

why care about Quantum estimation so as

I as I told you in the beginning in my

view uh Quantum estimation Theory or

Quantum information theory is just

speaking a little bit inside where this

poy given X comes from

okay so this P of Y given X which is the

you know the probabilistic description

of a noisy channel it's actually a uh it

comes from a collection of two things

you know y given X it comes from your

Quantum description of the information

bearing I'm saying like it doesn't have

to be light it matter so Quantum

description of the parameter that you

care about

and a povm description of the

measurement that you are using to

generate an estimate of that parameter

so this P of I given X the stress of rho

of x times pi y this is the povm

operator corresponding to the wire

outcome now all of classical estimation

Theory from say van trees book it it

deals with POI given X and Quantum

estimation Theory we take a peek inside

and we say well rho of X is something

that we can write down because we

understand how to write down the quantum

state of information bearing light

but we oftentimes we don't want to make

a guess on this measurement and Quantum

estimation Theory tools like the quantum

fissure information it can give you

official information like bound without

even having to talk about that measurement

measurement

and it automatically optimizes over all

possible measurement choices and this is

a common feature across all of these

different Quantum information Quantum

estimation bound the whole ever bound

the the quantum criminal bound the

hellstrom Bound for minimum probability

of error discrimination and so forth

so Quantum estimation Theory again that

the fisherian point of view where you

don't have any priors the Fischer

information uh is is replaced by the

quantum fission information I'm not

putting the bias term here but you you

there is a corresponding bias term you

can put there

this K of x

you want to write something like the

classical facial information you have

seen derivations that Carl showed you

derivation for example but if you look

at that quantity that D DX of P of I

given X you want something that um

that replaces this thing this log of P

and if you look at this thing uh you

know it is uh what is it it's a log so

so this thing um

if you look at that that quantity that L

of rho of x squared

this thing is just a Quantum version of

something you would directly write

instead of the probability distribution

would write with respect to the row

you write this operator L of L of rho of

X such that it is if I were to take this

on the other side

okay if I wanted to have a square of

so this squared is the con is this thing

is what I want to replace by this L of

rho of x

and because I cannot divide by a density

operator I take the d DX of the P of I

given X I have a d d x of rho X on this

side and the left hand side instead of

being just a multiplication by the

probability for technical reasons you

have to symmetrize this thing and write

this symmetric logarithmic derivative

but then it can actually solve for the

sld for the symmetric logarithmic

derivative operator that appears here

that takes the role of this score function

function

and you can write this in this very

interesting way

um and I think say you had derived this

um it goes as e to the minus rho of x

times Z this is the derivative of rho of

X with respect to X and then again e to

the minus rho f x of Z so if you look at

this the quantum pressure information I

never even specified the measurement choice

choice

but then

the point of the qfi is that this K of X

will be always larger than equal to J of

x no matter what measurement you pick

so it is an upper bound to the classical

efficient information and not just that

if you pick a measurement your

measurement this pie y to be a

projective measurement that is defined

by the eigen basis of this sld operator

which is implicitly defined by this

equation or explicitly defined by that

equation and plug it in here you will

get a p of Y given X for which if you

were to evaluate this equation

you will get K of X here

meaning you can achieve the quantum

fissure information

for a measurement Choice which is given

by the eigen basis of the sld operator

and you have again heard this previously

but I just wanted to instill this

important thing

and another thing to know is that this

measurement choice is not the only

measurement that will achieve the qfi

there are other measurements that will

also achieve the qfis you will see in

some examples I'll talk about in passive

Imaging contexts

now just like classical there is also a

Quantum version of the um the Bayesian

estimation Theory

and in Bayesian estimation Theory what

happens that you want to minimize the

mean squared error just like classical

and in this case you can write down an

explicit expression for the mmse

and you define an operator this operator

B that looks like the sld operator look

here instead of the row I have replaced

this by the average value of row with

respect to the probability distribution

P of X so this there are three operators

here gamma zero is the average of rho of

X over the P of X Gamma 1 and Gamma 2

are moment operators comma 1 is xpx rho

X DX and Gamma 2 is x squared e x row X DX

and this operator B its eigenvectors if

you were to choose this as the

measurement here

you will get a PO by given X which if

you go back and put into this expression

and evaluate the nmsc estimator and put

it back here you will get the minimum

possible mean squared error that quantum

mechanics allows you for the given row

of X and P of X and this expression was

derived by Supersonic in a paper in 1971

and actually transactions of information

Theory it's a beautiful result and it

again people have started looking at

multi-parameter versions of that and

there is a multi-parameter

Bayesian bound that was developed

recently but for a single scalar

parameter in principle if you solve this

operator equation

um to get these Gammas and then find B

from this equation you can put it here

into this formula and you can derive the

mmse and we have used that in the

context of passive Imaging that I'll

talk about in my next lecture okay so

that's all I'll call going to talk about

this though this slide is the overview

of all of estimation Theory tools but

let's remember this expression for

deficient information because we are

going to use that a little bit so let's

play with this estimation a little bit

more so as I said that we'll Define this

log likelihood function L of Y and

statisticians like to call the

derivative of that this score function

so this score function is you know

really if this function this is small it

means that that value of y is not as

sensitive to changes in x

uh let's massage this equation a little

bit more so as I Define the facial

information as the average value of this

L Prime operator in my previous slide

right so this is my definition of

sufficient information

so I'll take that definition and now

let's observe that if I were to look at

this um the P Prime in the numerator

here the L Prime is just P Prime over P

right so if you look at P Prime

um take the integral with respect to Y

and I'm going to assume I'm being sloppy

here I'm assuming that I can uh you know

move around the derivative and the

integral which can't always do but let's

say I can do that the derivative of this

P of I given X is obviously zero because

this integrates to one that's the

probability distribution similarly the

double derivative also is zero because

that integrates to one

so if I were to write the expected value

of this L Prime function where I treat y

as a random variable

this is the expectation now if I

substitute the value of L Prime from

here P Prime over p

the P cancels out and the P Prime

integral is 0 from here

right so I have the expected value of

this L Prime is 0 hence

the expected value of the squared of L

Prime is nothing but the variance of L

Prime right because the variance of a

random variable is the expected value of

the square of the random variable right

and then minus the square root of the

mean the mean is zero in this case

so this is another formula that will

sometimes come in handy so let's

remember that as well that's the second

way to write the official information

and I'm going to derive a third way to

write the official information and I'll

continue after that two examples

um so let's do a little bit more math

here a very simple math I'm going to

take that L of Y given X that L Prime

function that I defined earlier so this

is L Prime remember L Prime is p Prime

over p

let's let's take a derivative of that

one more derivative of that

so P Prime over P 5 derivative with

respect to x what will I get you apply

the U by V rule

okay so p double Prime P minus P Prime

squared over P of Y given x squared and

you get a p Prime over P from the first

term minus this is just L Prime squared

now if you take the expected value on

both sides the right hand side I take

this whole thing I take an expectation

with respect to this P or Y given X distribution

distribution

um and if you look at the first term the

P cancels out you get a p Prime integral

and I just showed in the previous slide

that this is zero right this integral is zero

zero

so you just get this this thing left so

what we just derived is a j facial

information is the negative expectation

of the second derivative of the score of

the uh of the L of Y given X the

likelihood function

so this gives us the yet the third way

of writing the official information

and depending upon the problem context

we are looking at you can use the first

the second or the third wave they're all

the same thing

okay so just this is just a summary so

if I give you a p of Y given X if I tell

you to derive the efficient information

you can either take the expectation of

these of the square of this code

function take the variance of that L

prime or the negative expectation of L

double Prime all of them are the same thing

thing

all right so uh what if

X was not the quantity you were

interested in and we will see some

examples in Quantum Metrology uh where

we will see this happen X is something

that is your parameter that gives rise

to Y but what you really care about is

uh is this new parameter Z such that X

is a function of Z so F let's assume

it's a differentiable function but it is

z that you actually care about

estimating so you want to estimate Z so

this is that's why I'm calling it Z hat

of Y but you know the functional

relationship between x and z and you can

invert it to write z as some f inverse

of x

have you heard of jacobians in calculus

when you you probably have to use that

in a different context but it's

basically the idea of Jacobian so you

can write down the Fischer information

for Z from the fission information of X

with X substituted at F of Z but you

need this additional correction term

which is the squared of the F Prime of C

and we will see we'll actually use it in

some examples that we will derive

together okay so let's remember this uh

that how you estimate a function of the parameter

parameter

okay so let's do a couple of examples so

the first example I'll do I think you

heard this example both in Carl's talk

and I believe in size talk as well so I

will do an example where we will look at um

say you have a box so I wrote this box

where X gives rise to y

let me say that y Only Takes Two values

okay so Y is either one or zero

and it is 1 with uh with a probability X

and say it is 0 with probability 1 minus X

so X in this case the parameter that you

care about is nothing but the

probability of Y taking the value 1 or 0.

I want to find the facial information

you want to try this calculation

okay so let's let's say that let's start

with writing a p of Y given X

any ideas on how we might compactly

write this probability distribution of Y

given X

you can do it in many different ways but

any any ideas of how you might try to do

that such that because you have to take

this derivatives and so so on to

calculate these the facial information

so I have to first have a POI given X in

order to go from here to L Prime and all that

when

cosine function so what what cosine

it might work what what do you have in mind

so how would you fold in X into that so

you need something such that this is

going to give you X or 1 minus X if I

substitute x equal to

um if the value of x I substitute I want

to get 1 or 0 depending upon whether uh

you know with the right probability

so let me give you actually there are

many functions that work let me pick a

simple one that you know that is easy to

work with so let's look at that so x to

the Y times 1 minus x to the 1 minus y

okay so why does this work so if you

just stare here now if I put in the y

equal to 1 it's a very trivial way if I

put y equal to 1 I get X this becomes 1

if I put y equal to zero

but what happens I get 1 minus X so very

simple one okay so in this case I we

have to pick our one way toward the

other to derive it so we'll take the log

likelihood function so log of P of y k 1

X log multiply so y log X Plus 1 minus y

log 1 minus X

and uh then L Prime can just take a

derivative it's very easy L double Prime

you can also differentiate it one more

time and here I'm just using the Third

Way of calculating the official

information it's the negative

expectation of the second derivative of L

and uh if you take this if this thing

you take the expectation of this

how do you take the expectation of this

so if you want to take an expectation of

this you have to take an expectation

over that probability distribution right

so what you will want to do is that with

probability X you are going to put in y

equal to 1 meaning you will say y equal

to 1 you put here and then multiply that

by X then put y equal to 0 multiplied by

1 minus X solve couple of lines and you

will get this answer 1 over x times 1

minus X okay

so yes you should try this this is a

very simple example and this example is

what we are going to use to derive the

Heisenberg limit and some very very

interesting examples of quantum

Metrology calculations we will do

probably the simplest example the next

one again I would like you to remember

this let me write this formula down so

that we can use it in the future

calculation so the J of X here is 1 over

x times 1 minus X

the other one is a gaussian distribution

so let's say Y is a gaussian comes from

a gaussian distribution whose mean is

unknown and the variance is Sigma

squared now if you were to calculate the

uh the J of x from this you will take

the likelihood function and this here

I'm using the same formula minus

expectation of the L double Prime I get

a 1 over Sigma squared now let's look at

it and see sense

sense

if the gaussian distribution has an

unknown mean and you are getting a

random sample from that if the variance

of the distribution is very small then

that y will give you X the information

about X the mean value much better right

than if it's a fat distribution with a

big variance so it makes sense that if

we have a small variance your facial

information will be high about X

but you can calculate this also very easily

easily

the third one that this that is also so

this one will not use but it's good to

interesting to note that what if the

mean is known and the variance is unknown

unknown

and here you can calculate the official

information comes out at 1 over 2x

squared again it goes as one over this

the variance

now the square of the variance in this

case but the main difference you see

between these two is that the efficient

information in this case is a function

of the parameter itself

this is another annoying thing about

fishery and estimation Theory which we

have to live with unfortunately that the

estimation Precision about a parameter

is oftentimes dependent on the parameter

itself so the true value of the

parameter also determines how well you

can estimate it but not always there are

example situations where it doesn't

appear in G of x

this one I am going to use I am going to

write down so it's a poisson

distribution have you heard of the

poisson distribution the poisson

distribution and x y takes values over

integers and X is a scalar real number

between 0 and infinity and I get a

random sample y I want to estimate the

mean of the distribution X and in this

case J of X is given by 1 by X so my P

of Y given X is is a poisson

distribution it's e to the minus x x to

the Y over y factorial

and J of X for this is given by 1 over X

so we'll be needing this in some examples

examples

uh I suggest you try out all these

examples yourself they're very very easy

you can take a picture or these slides

are these are recorded so you can

download them and work these examples out

out

okay so a couple of important points

about estimators I talked about Fisher

information you know how to calculate

pressure information from P or Y given X

the whole mechanism is there

I told you about the maximum likelihood

estimator I told you about the mmsc

estimator the minimum means quad

estimator which is the expected value of

x squared y so if I give you P of Y

given X give you P of X also you

calculate P of x given Y and its

expectation is your optimal estimator

for Bayesian

there is no P of X right that's the main

difference between fishery and Bayesian

and here the maximum likelihood

estimator is simply the picking the

value of x for which the likelihood

function is the highest

for Bayesian estimation Theory there is

also a much easier estimator that people

often use because this integral is

sometimes hard to calculate or the mmsc estimator

estimator

what if I use the P of X information for

Bayesian you have to somehow use the

information for the prior right so what

you can do is that just like the maximum

likelihood estimator you pick the value

of x for which the probability

distribution highest but instead of

picking the value of x for which P of Y

given X is the highest you pick the

value of x for which is the joint

distribution is the highest

and this is called the map estimator or

the maximum a posteriori probability estimator

estimator

now what you can show is that so by the

way so mmsc obviously is the best

because it achieves the minimum mean

squared error but the map estimator is

easier much easier to write down if I

get a value of y you can go ahead and

look up a probability distribution you

don't have to do any base rule you don't

have to do any integrals

but the very interesting thing is that

when n becomes large

the mean squared error for all of these

three estimators they all approach the

expected value of the one over the

efficient information

under the prior probability distribution

of X the Bayesian case this is all Bayesian

Bayesian

so even for the Bayesian case if you

wanted to use ml meaning you completely

ignored the prior or you used uh map or

mmse doesn't matter which one you choose

to use all of them their MSC will

saturate to the expected value of one of

the official information

and what is it saying why is this true

well the reason this is happening is

that as n goes to Infinity you're

probably your your conditional

distribution of the true value of the

parameter given the measurement observed

that you're observing

converges to a gaussian distribution

whose mean is the true unknown value of

the of the parameter uh its average

value under the distribution P of X with

a variance given by 1 over n times G of

X 0. so what's happening is that the

distribution of P of X as you are

getting more and more and more samples

is like a gaussian that is doing this it

is moving like this with the mean going

to x0 and the variance also shrinking as

1 over n times 0 of x 0 add the facial

information value evaluated at that true

value of x0

so that is why when n becomes large you

can completely ignore the the prior and

it will still work so let's do c One X

this is one example so this by the way

this expectation of the one over J of x

uh this people often call the expected

Crema Rao bound or the ecrb so I talked

about the bcrb remember that was the

potentially loose lower bound to the

minimum mean squared error and this is

the expected criminal bound which is

tracked by the maximum likelihood estimator

estimator

so this is one example I stole from Van

trees book and by the way if you are if

you're looking for a textbook for In

classical estimation Theory this is my

favorite uh there's many many good books

but I learned at least as a graduate

student undergraduate student all of my

estimation Theory from bantry's book so

in this example he takes a poisson

distribution just like I wrote over here

and he took this crazy prior

distribution so this is a generalization

of the exponential distribution but with

these parameters Alpha and beta or a

alpha and B when you set Alpha to 1 and

B to 1 by mu you get a the person the

the uh sorry the uh exponential distribution

distribution

all right so what he plots here is this

is the number of samples

and on the y-axis he's plotting the root

mean squared error the mean squared

error with a square root so see what is

happening the stars are the simulated

value of Maximum likelihood so you are

never you are not even using the prior

information and as expected the ecrb is

tracking the ml estimator's performance

and when n becomes High

all of these estimated ecrb and the and

the map and the mmsc estimators

performance which is the circles and

these pluses they all converge to the

same thing which is given by the ecrb

which we saw in the previous slide right

and then if you look at this black line

this is the actual mean Square the

minimum mean squared error which they're

numerically computed and the pluses are

the simulations and see the map

estimator even though it is a much

easier to calculate than mmsc it is only

a little bit worse than an MSC so map

estimator is pretty good it is it is

much easier to calculate that nms you

don't have to do this integral and the

inversion to get P of x given y

but this is that dcrb that Bayesian

creamer outbound is that I that I talked

about uh and see that this is a loose

bound it's it's a bound to the mmsc but

it's not a good Bound for in this case

uh because there is no estimator that

will attain it the minimum mean squatter

is above it

all right so uh for fishering and

Quantum estimation Theory I'm only going

to make a few notes because you have

heard a lot of it from Carl's talk and

size lecture a couple of notes uh the

sld eigenvectors attain the qfi and

whenever I say attain the qfi what I

actually mean is that if you take that

sld operator the L operator I shown

calculate its eigenvectors and use that

as a projective measurement to measure

rho of x you get a PO by given X

calculate the classical official

information G of x from it it will equal

the quantum official information K of X

that's what I mean by attaining the fish

the quantum facial information

so it's CFI of this measurement equals

the qfi

and the sld eigenvector measurement

could depend on the parameter itself and

this is again a funny thing that happens

with the sld measurement the sld

measurements eigenvectors themselves can

depend upon the parameter so how do you

deal with it then so there is a very

recent paper that addresses this issue

so if I have n copies of my density

operator my state that I want to want to measure

measure

so what you do is that you calculate

let's say the sld operator

and you want to you know make start

making measurements so what you do is

that you take the first square root of n

copies of the density operator

and you measure it with absolutely any

measurement you want that has a positive

classical facial information

use that measurements output to get an

estimate a pre-estimate of the parameter X

take that pre-estimate and plug that

into your sld eigenvectors expression

which can depend on X and use that as LD

eigenvector for measuring the remaining

n minus square root of n copies of rho

and you can prove that as n goes to Infinity

Infinity

even though your sld eigenvectors

project your eigenvectors dependent on X

that you will end up achieving the qfi

for that prior unknown value of x so

this is the two-stage measurement that

was developed by um

a few years back so very very

interesting result

okay so another thing I mentioned or I

had also mentioned this qfi attending

measurements are not unique uh and this

expression is something I'm going to use

a couple of times so I will not derive

it but I will urge you to try to derive

this if your row of X is a pure State

meaning if rho of X is PSI X gets PSI X

bro uh you can actually write a very

nice compact expression for the

efficient information

um in terms of the inner product of the

derivative of the PSI x with respect to

X and the derivative magnitude squared

of PSI and PSI dot inside

we'll be using it in some calculations

you will do

okay so let's now take a break from

estimation Theory we will start looking

now at uh the the quantum description of

light and then we'll we'll go from there

no my my research group when my students

go on for a hike they have this uh

they're obsessed with this thing I don't

know why they do this they look for the

perfect size shaped Cactus and they take

a picture of that

and we have this repository in our group

Google Drive of all these PSI shape

campuses so

but I don't know if this is where this

ranks in our in that list

all right so let's start with a pulse of

laser light

um I'm going to assume no quantum

mechanics here I know you all have

different various levels of quantum

mechanics this is the simplest picture

of a laser light pulse

um where I first let's say Define a mode

a mode is just like it's a shape it's a

shape of a field it's like a bucket in

which you put some water so I Define a

mode this is a flat top mode and the

mode functions always will be normalized

to one so magnitude squared of Phi of T

over T from 0 to T is one

once I Define a mode

the only thing you need to just to

specify to fully specify the quantum

state of an ideal laser light pulse

are two things an amplitude complex

number Alpha

and a phase Theta which is a phase

reference with respect to some chosen

phase reference like a phase from from

chosen reference so Alpha

so sorry it's Alpha meaning a real

number and that phase so that's the

whole thing is a complex number Alpha so

square root of n times e to the J Phi

here I'm just showing schematically a

coherent State and in this thing what is

buried I'm not showing is the the

oscillatory portion so there's a over

here the mode shape you'll have to

multiply by some e to the J Omega naught

T so this is the oscillatory portion if

that doesn't change I don't have to

worry about it writing it in this blue

thing I'm showing this is just the noise

ball around the oscillatory component so

I try to measure a quadrature I'll get

some noise I'll come to that in a second

but keep this picture in mind with that

picture again if I specify a mode the

only thing you need to specify for a

laser light console is one complex

number Alpha and in Quantum Optics we

call that a coherent state of that mode

in Quantum Optics you the most General

way to talk about light is first

described first identifying what setup

orthogonal modes you want to talk about

the three degrees of freedom space time

and polarization or space frequency and

polarization time and frequency are just

too conjugate of each other

so one should Define orthogonal buckets

in this three variables then you can you

have to talk about the quantum state of

this collection of modes if you're

dealing only with laser light all you

have to do is to specify one complex

number for each one of those buckets

if you're talking about statistical light

light

meaning the normal life for example then

you have to describe not just those

complex number for each of those buckets

but a probability distribution over

those complex numbers there's this whole

field of statistical Optics that deals

with coherence Theory and all they are

doing is this problem they're playing

this probability distribution over those

complex numbers over orthogonal modes

okay so that's the coherent State and if

I were to detect a coherent state with

an ideal Photon counting detector and

infinite bandwidth ideal no dark click

Unity Quantum efficiency Photon detector

I'm going to get a poisson point process

what is a poisson point process okay

okay um

how many of you have heard about a point

process before if you have taken any

course on queuing query perhaps um

probably not okay so let me describe

what a person point process is it's a

very simple concept you look at so you

start observing your light pulse starts here

here

uh if you were to feed this into an

ideal detector you will see arrivals or

Photon clicks if you want to call them

that arriving at a rate given by Lambda

which is the magnitude squared of these

of the field which in this case I'm just

taking to be a flat top it is just e

squared where e is the value of the

field over here okay

and uh constant rate but it also has

this property that if I were to slice

this time into tiny little segments

let's say of segment length Delta say

Tau then for every segment I'm going to

have a probability of let's say this is

my 0 to T

and I like look at a small segment town

the probability of there being a click

or an arrival in this uh

segment will be Lambda times Tau and

probability of no click

and another thing is that the

probability of a click appearing in the

next time bin will be independent of the

probability of actually whether click

occurred or not in the previous timeline

so that's all so every little time then

you have a probability of a click or not

click but it's an independent from bin

to bin

that's all you need to derive the fact

that if you count the number of clicks

over the entire duration of the pulse it

will have a poisson distribution with a

mean given by the mean photon number

which is the integral of the rate over

the time so this is the photons per

second clicks per second integrated over

the time so this is the unit of photon

so n is e Square t and n is in this

notation it's magnitude Alpha squared

okay so that's the poisson point process

so this is as you can see with the

poisson point process your your rate of

arrival is driven by your squared

intensity in photon units of the

coherent state but it's a random process

meaning if I were to prepare identical

laser light pulses and did this

experiment each time I'll get a

different click pattern but each time

the click pattern will come from that

poisson point process

so oftentimes I'm going to use this

notation a coherence State Alpha going

into a detector generating a random

variable that has a poisson distribution

with a mean magnitude Alpha squared

now this measurement uh there's no way

for me to measure the phase five right

because no matter what Phi is I'm always

going to get the poisson point process

with the same exact rate so this direct

detection cannot measure the phase of alpha

alpha

to break the measurement of phase of

alpha there are two things I must

introduce so the one is interference of

when you take a beam splitter a beam

splitter looks like it's a piece of

glass okay bulk beam splitter it's a

cube you can also buy fiber beam

Splitters that look like couplers two

fibers come together and then they meet

each other and then they go out okay so

beam splitter has two inputs and two

outputs is defined by two quantities one

is transmissivity the other one is a

phase so I'm going to take

ETA as the transmissivity which is the

fraction of input energy that I put at

this input that goes out to this output

meaning if the input has some energy eat

a fraction of this energy will go this

way and one minus Theta fraction of the

energy from this input will go this way

and phase is our number between 0 to 2 pi

a beautiful property of coherent States

is that if I were to interfere two

coherent States in a beam splitter

at the output I get two coherent states

that are completely independent of each other

other

um in some tensor product cohere instead

and their amplitudes are given by this

equation so this is a unitary matrix

multiplication right so beta 1 beta 2 is

multiplied by alpha 1 Alpha two as a

column Vector with this unitary Matrix

so the convention I'm going to stick to

is this convention for the later calculations

calculations

um so for example if I were to take this

example here Alpha One is alpha alpha 2

is minus Alpha and let's say Phi is 0

ETA is square root of one half meaning

it's a 50 50 beam splitter and I get an alpha

alpha

plus minus Alpha by root 2

over here which is 0 and Alpha minus

minus Alpha over root 2 which is square

root of 2 Alpha okay

so this is uh just showing how you can

get fully constructed versus fully

destructive interference this is well

known how you can interfere two laser

light pulses

you can also do more fancy things for

example what if I choose Alpha 2 to be

some complex number beta divided by

square root of 1 minus ETA

and the face to be zero and the amp the

X the transmissivity I take it to we say

99.9 something very close to one

then if you were to substitute this into

this first equation let's see what we

get take this this is Alpha two let's

put Alpha 2 over here

the square root of 1 minus Theta cancels

it's if I put Alpha 2 here Phi is

already 0 so this whole term becomes beta

beta

and the first term alpha 1 is Alpha but

ETA is very close to 1. so I get

something that is close enough to Alpha

plus beta by choosing ETA to be very

close to 1. so this setup can be used to

add two complex numbers I can add Alpha

to Beta And I can get Alpha plus beta

and this is something in tomorrow's

lecture you will see how I'll use it to

design better receivers for State

discrimination and this is something

that was first uh studied in the context

of optical State discrimination by by

Kennedy by many years ago 1970s and then

his former student Sam dolinar who did

some very very interesting work on using

this just this complex number addition

and Photon detection to come up with the

quantum optimal receiver to teleport

between two states of laser light pulses

okay so with that

um I'm gonna just you know make a point

that if you were to do this displacement

or this edition of a complex number

prior to detection

then you still get a poisson outcome

except that its mean has now changed to

Alpha plus beta magnitude squared so by

doing something before you detect the

information bearing light you can change

the detection basis and and and such

changes to the detection basis is

something you can has have as a tool to

uh for for use for maximizing the

information extraction efficiency and

again we will see examples of that in

tomorrow's lecture

all right so what if you were interested

in estimating the phase of the field

then what you do is um you can do

homemodyne detection so the way homodyne

detection works is that you take that

coherent State Alpha

and you interfere that with another

coherent state that is locally prepared

and this coherent state is a very strong

coherence State Alpha Lo

but see the beam splitter here is a 50

50 beam splitter

so what you get on these two outputs are

two coherent States uh can you tell me

what are the coherent States you're

going to get are the two outputs

so you have Alpha and Alpha yellow and

it's a 50 50 beam splitter so if you

remember the convention that I that I

wrote down the first output will have

Alpha plus Alpha Lo over root 2 right

and the second one is going to have

Alpha minus Alpha Lo over root 2. and

what if I now do Photon detection that

normal Photon detection that will be

your poisson outcome on both of those

outputs so what will happen is that you

will get two random variables K1 from

the first detector which will be a

poisson random variable

whose mean is magnitude squared of alpha

plus Alpha Lo over root 2.

and the second one will be a poisson

random variable

with a mean of alpha minus Alpha L over 2.

magnitude squared

and then I will take you know this that

for a poisson distribution

if you take the poisson distribution

and make the mean value very very large

it approaches the gaussian or the normal

distribution so it goes to the normal

distribution with that same mean and

because the poisson has the variance

equals the mean the variance of the

gaussian is also Lambda

so if you were to take K1 and K2 and

pretend that they are gaussian

distributions why can you pretend that

their gaussian distributions because

this Alpha no has chosen to be very

large the mod squared of that is Lambda

is very large for both of those detectors

detectors

and if you write this as K and you you

scale it appropriately you can prove

this or gaussian minus a gaussian will

get a convolution of two gaussians also

a gaussian its mean value you can prove

will come out to be a this is the two

lines of a calculation you should try to

do yourself is equal to the real part of

alpha times e to the E uh say this is I

and J are the same thing I'm an

electrical engineer I still oftentimes

would write J for the I because in

electrical engineering they they reserve

the letter i for current

but it's the same thing it's the square

root of minus one

so Alpha e to the J Theta L over Theta l

o is the phase of the local oscillator

with respect to the input Alpha and the

variance will come out to be one fourth

now this is a very interesting thing

that with homodyne detection I can get a

measurement whose mean value is any

chosen quadrature of that complex number

meaning if I choose the Theta Lo such

that I can measure the real part of

alpha I can measure the imaginary part

of alpha I can measure any quadrature in

between real and imaginary

but the price I pay is that no matter

what I'm trying to measure

I will always have this one by four at

the variance

and this variance sometimes in the

classical world people will call that

short noise of the local oscillator

okay but in the quantum language there

is nothing but the quantum limited noise

of a single of the quadrature operator so this homogene detection measurement

so this homogene detection measurement on a coherent state is something that

on a coherent state is something that people often use to measure phase

people often use to measure phase information both in Communications and

information both in Communications and sensing

sensing all right so now let's go a little bit

all right so now let's go a little bit more Quantum so I talked to you about

more Quantum so I talked to you about the mode and the coherent state of a

the mode and the coherent state of a mode

mode there's something you can talk about is

there's something you can talk about is the number state of a mode or a fox

the number state of a mode or a fox state of a mode so remember when I said

state of a mode so remember when I said a coherent state of a mode I said that

a coherent state of a mode I said that if you were to detect photons in it you

if you were to detect photons in it you will get a poisson point process you get

will get a poisson point process you get a random number of photons with a rate

a random number of photons with a rate that is given by the the in the

that is given by the the in the intensity they pulse

intensity they pulse but in a number state of a mode if I

but in a number state of a mode if I give you that same mode that flat top

give you that same mode that flat top laptop mode like that but I say I am

laptop mode like that but I say I am exciting that in the fox say one

exciting that in the fox say one what does that mean if I prepare that

what does that mean if I prepare that and put it onto a detector

and put it onto a detector I'll get one click exactly one click no

I'll get one click exactly one click no probability

probability where the click will come

where the click will come during that zero to T that will be

during that zero to T that will be probabilistic that will be given by the

probabilistic that will be given by the pro the probability distribution will be

pro the probability distribution will be the the integral of the the mode shape

the the integral of the the mode shape which integrates to one magnitude of Phi

which integrates to one magnitude of Phi t squared

t squared but but they will be exactly one photon

but but they will be exactly one photon now if I give you a two Photon Fox State

now if I give you a two Photon Fox State um prepare that and I detect photons

um prepare that and I detect photons with it you'll get exactly two clicks

with it you'll get exactly two clicks every time you do an experiment

every time you do an experiment this time these two clicks don't come

this time these two clicks don't come from a poisson point process they don't

from a poisson point process they don't come up from a memory less process in

come up from a memory less process in fact there is a there's a process

fact there is a there's a process underlying that random arrival of these

underlying that random arrival of these photons which I will not have time to

photons which I will not have time to discuss today that Horus un and Jeff

discuss today that Horus un and Jeff ship here and I believe Carl had done

ship here and I believe Carl had done also some work on that writing down this

also some work on that writing down this um

um Markov or Stat process description of

Markov or Stat process description of arrival of photons for screen slide and

arrival of photons for screen slide and number States but the main thing to

number States but the main thing to remember here is that it said exactly

remember here is that it said exactly the number of photons at the end that's

the number of photons at the end that's the definition of a fox state

the definition of a fox state okay so

okay so um Fox States they form a complete

um Fox States they form a complete orthonormal basis for any state of a

orthonormal basis for any state of a bosonic model of an optical mode and a

bosonic model of an optical mode and a simple conceptual way to think about it

simple conceptual way to think about it is that if I give you two states two Fox

is that if I give you two states two Fox States you can tell a part between those

States you can tell a part between those two Fox states with probability of error

two Fox states with probability of error Zero by simply doing a photon

Zero by simply doing a photon measurement because you'll get a number

measurement because you'll get a number of clicks on you that will tell you what

of clicks on you that will tell you what fox State you had

fox State you had and the only way you can have two states

and the only way you can have two states that you can discriminate with

that you can discriminate with probability of error zero is if they are

probability of error zero is if they are orthogonal states so all the foxtes are

orthogonal states so all the foxtes are orthogonal and they span all the photon

orthogonal and they span all the photon numbers so they must span all the states

numbers so they must span all the states that uh of a bosonic mode where this is

that uh of a bosonic mode where this is being sloppy but this is a good way to

being sloppy but this is a good way to understand why Fox states form a basis

understand why Fox states form a basis for any state

for any state so a coherent State can also be written

so a coherent State can also be written in The Fault basis in this following way

in The Fault basis in this following way and now you can see that you can

and now you can see that you can interpret Photon detection that I talked

interpret Photon detection that I talked about as a measurement of the coherent

about as a measurement of the coherent state in the fog basis

state in the fog basis and as you know from quantum mechanics

and as you know from quantum mechanics the probability of the outcome n is the

the probability of the outcome n is the magnitude squared of the fox State n

magnitude squared of the fox State n with the coherent State magnitude

with the coherent State magnitude squared which is this poisson term e to

squared which is this poisson term e to the minus n n to the N Over N factorial

now uh if you were to apply a small phase to a coherent state

phase to a coherent state phase can be applied by a small time

phase can be applied by a small time delay for example in an optical fiber or

delay for example in an optical fiber or free space setting so if you apply a

free space setting so if you apply a phase Theta on a coherent State you get

phase Theta on a coherent State you get Alpha times e to the I Theta you saw

Alpha times e to the I Theta you saw that in the description of the beam

that in the description of the beam splitter I showed you this is a phases

splitter I showed you this is a phases simply the special case of the beam

simply the special case of the beam splitter when ETA equal to one

splitter when ETA equal to one if your transmissivity was one there was

if your transmissivity was one there was only one phase this is what you will get

only one phase this is what you will get if you apply a phase to a fox state

if you apply a phase to a fox state would you still get a fox statement with

would you still get a fox statement with the face outside as a as a term that is

the face outside as a as a term that is irrelevant if you just had a single Fox

irrelevant if you just had a single Fox state but if you had a state where you

state but if you had a state where you had superpositions of foxtails then this

had superpositions of foxtails then this phase plays an important role you don't

phase plays an important role you don't have to worry about the a and a dagger

have to worry about the a and a dagger operators because I don't know how many

operators because I don't know how many of you are familiar with Quantum Optics

of you are familiar with Quantum Optics and we will not need to use these

and we will not need to use these Annihilation creation operators in in at

Annihilation creation operators in in at least this these two lectures

least this these two lectures okay so what is a squeeze state so you

okay so what is a squeeze state so you have again seen these uh pictures before

have again seen these uh pictures before uh but now you can understand squeeze

uh but now you can understand squeeze state in terms of what it does in a

state in terms of what it does in a homody and receiver so let's say take

homody and receiver so let's say take that same homody and receiver that I

that same homody and receiver that I showed you but instead of a coherent

showed you but instead of a coherent State Alpha I put in something that is

State Alpha I put in something that is called a squeeze State

called a squeeze State um and if you remember for the coherence

um and if you remember for the coherence State I showed this noise ball with this

State I showed this noise ball with this one fourth and one-fourth so no matter

one fourth and one-fourth so no matter which quadrature you measure you get

which quadrature you measure you get this exactly the same noise variance for

this exactly the same noise variance for a squeeze state that noise variance uh

a squeeze state that noise variance uh is different depending on which

is different depending on which quadrature you are measuring so your

quadrature you are measuring so your output is still a gaussian distribution

output is still a gaussian distribution with the mean equal to the center that

with the mean equal to the center that complex number Alpha but your variance

complex number Alpha but your variance sweeps anywhere between 1 4 times e to

sweeps anywhere between 1 4 times e to the minus 2 R to 1 4 time e to the two R

the minus 2 R to 1 4 time e to the two R where R is often termed the squeezing

where R is often termed the squeezing parameter

parameter so this

so this um this PSI is the complex quizzing

um this PSI is the complex quizzing parameter whose real part is the amount

parameter whose real part is the amount of squeezing

of squeezing and it's a phase of this is the

and it's a phase of this is the direction of squeezing so if I take this

direction of squeezing so if I take this ellipse and turn it like that that will

ellipse and turn it like that that will be changing Theta

be changing Theta so now you have two complex numbers

so now you have two complex numbers Alpha being the center of the ball

Alpha being the center of the ball and this is I which tells you how much

and this is I which tells you how much to squeeze

to squeeze in which direction the squeezing is in

in which direction the squeezing is in okay so that's a squeeze state

okay so that's a squeeze state and if you look at in the time domain

and if you look at in the time domain what the squeezing means is that

what the squeezing means is that depending upon which quadrature you are

depending upon which quadrature you are measuring the noise balls size can

measuring the noise balls size can change

change and you can squeeze it in different

and you can squeeze it in different directions you can squeeze it this way

directions you can squeeze it this way you can squeeze it this way this is

you can squeeze it this way this is called phase quizzing this is called

called phase quizzing this is called amplitude squeezing and the noise ball

amplitude squeezing and the noise ball where it amplifies and where it shrinks

where it amplifies and where it shrinks it that changes

it that changes okay so

okay so um another important thing to remember

um another important thing to remember is that uh squeeze state for coherent

is that uh squeeze state for coherent State the mean photon number was just

State the mean photon number was just this magnitude squared of the center of

this magnitude squared of the center of the ball

the ball for a squeeze State there are two

for a squeeze State there are two contributions the mean photon number one

contributions the mean photon number one comes from magnitude Alpha Square which

comes from magnitude Alpha Square which is still the magnitude squared of that

is still the magnitude squared of that complex number which is the center of

complex number which is the center of the ball at ellipse but plus there is an

the ball at ellipse but plus there is an additional term which is a sine

additional term which is a sine hyperbolic squared of this R parameter

hyperbolic squared of this R parameter so there are two contributors so now

so there are two contributors so now even if Alpha is zero even if my ellipse

even if Alpha is zero even if my ellipse is centered over here in the origin that

is centered over here in the origin that squeeze state which is known as the

squeeze state which is known as the squeezed vacuum still has photons in it

squeezed vacuum still has photons in it sine hyperbolic Square R whereas a

sine hyperbolic Square R whereas a coherent state in the middle has no

coherent state in the middle has no photons even though it still has to have

photons even though it still has to have the one over four variance on both

the one over four variance on both quadratures so even a vacuum state if

quadratures so even a vacuum state if you were to do homonym you will get a

you were to do homonym you will get a zero mean gaussian variable but variance

zero mean gaussian variable but variance of 1 4. but it has no photons in it

of 1 4. but it has no photons in it okay so for a squeezed vacuum State uh

okay so for a squeezed vacuum State uh if you were to write this down in the

if you were to write this down in the fog basis uh what you what interesting

fog basis uh what you what interesting thing that you see is that you only get

thing that you see is that you only get contributions in the even photon number

contributions in the even photon number terms and I will not be deriving this

terms and I will not be deriving this expression but but the main thing I want

expression but but the main thing I want you to take home is that first squeeze

you to take home is that first squeeze vacuum has photons in it and then it has

vacuum has photons in it and then it has this non-personian Statistics it's very

this non-personian Statistics it's very weird statistics so you only have zero

weird statistics so you only have zero two four six photons and so forth

two four six photons and so forth um in the time domain if you want to

um in the time domain if you want to visualize vacuum it is nothing right so

visualize vacuum it is nothing right so there is no oscillation in a vacuum

there is no oscillation in a vacuum adhesive System Flat so that you if that

adhesive System Flat so that you if that the one-fourth noise ball is always

the one-fourth noise ball is always there but there is no notion of a

there but there is no notion of a oscillatory component even it's just

oscillatory component even it's just vacuum is vacuum

vacuum is vacuum but notice in squeezed vacuum

but notice in squeezed vacuum it is still vacuum

it is still vacuum okay so there is no

okay so there is no oscillatory component that does this

oscillatory component that does this like a coherent state but there is still

like a coherent state but there is still an oscillatory component with a given

an oscillatory component with a given Center frequency so squeezed vacuum

Center frequency so squeezed vacuum you will have to say squeezed vacuum at

you will have to say squeezed vacuum at 1550 nanometers or squeezed vacuum at

1550 nanometers or squeezed vacuum at 780 nanometers so that determines

780 nanometers so that determines the the frequency with which you see

the the frequency with which you see this uh you know that the quantitarians

this uh you know that the quantitarians go up and come down back from this mine

go up and come down back from this mine one over four e to the minus two out one

one over four e to the minus two out one over four e to the two r

over four e to the two r and this is a coherence State this is a

and this is a coherence State this is a phase squeeze and amplitude squeeze

phase squeeze and amplitude squeeze State all right so how do you generate

State all right so how do you generate squeeze vacuum well what you do in the

squeeze vacuum well what you do in the lab is that you squeeze the vacuum to

lab is that you squeeze the vacuum to take the vacuum State and then you pass

take the vacuum State and then you pass it through a chi to non-linear Optical

it through a chi to non-linear Optical process for example there are other ways

process for example there are other ways to do it as well and this thing is

to do it as well and this thing is called a squeezer

called a squeezer again I will not go into the derivation

again I will not go into the derivation of a squeezer in this talk but I'm happy

of a squeezer in this talk but I'm happy to describe to you those of you who are

to describe to you those of you who are inclined to understand how this process

inclined to understand how this process works but this is just a picture from my

works but this is just a picture from my lab my graduate student Alex vent has

lab my graduate student Alex vent has built a squeezer this is how it looks

built a squeezer this is how it looks like in the laboratory it's a he has a

like in the laboratory it's a he has a periodically full KTP crystals sitting

periodically full KTP crystals sitting in a uh in a in a bow tie cavity this is

in a uh in a in a bow tie cavity this is an oven to fine tune the face matching

an oven to fine tune the face matching conditions and this thing generates a

conditions and this thing generates a squeeze vacuum at a few DBS or squeezing

squeeze vacuum at a few DBS or squeezing at 15 50 nanometers

at 15 50 nanometers all right so why care about squeeze

all right so why care about squeeze vacuum so let's look at perhaps one of

vacuum so let's look at perhaps one of the earliest examples of use of squeeze

the earliest examples of use of squeeze vacuum so if I take a hormone detection

vacuum so if I take a hormone detection on a coherent State Alpha what did I say

on a coherent State Alpha what did I say let's say that Alpha is a real number I

let's say that Alpha is a real number I will get a gaussian distribution with a

will get a gaussian distribution with a mean of Alpha and a variance of

mean of Alpha and a variance of one-fourth so I told you about that

one-fourth so I told you about that so this was a

so this was a it's a very very simple observation

it's a very very simple observation I don't even remember who made this

I don't even remember who made this observation Carl you may remember was

observation Carl you may remember was this your

this your Horus one we wrote paper that that

Horus one we wrote paper that that injecting squeezing

injecting squeezing onto the vacuum Port of a beam splitter

onto the vacuum Port of a beam splitter you can recover the signal to noise

you can recover the signal to noise ratio of a hormone receiver as if this

ratio of a hormone receiver as if this beam splitter was never there

beam splitter was never there so what I mean by that is if I take a

so what I mean by that is if I take a homodyne detection on a coherent state

homodyne detection on a coherent state the signal to noise ratio is the squared

the signal to noise ratio is the squared of the mean divided by the variance it's

of the mean divided by the variance it's 4 times Alpha squared

4 times Alpha squared but so if I had a coherence State beta

but so if I had a coherence State beta if I pass it through some beams that are

if I pass it through some beams that are Kappa with vacuum here I would get a

Kappa with vacuum here I would get a signal to noise ratio that will be given

signal to noise ratio that will be given by four times Kappa times beta square

by four times Kappa times beta square right because the mean will be square

right because the mean will be square root of Kappa times beta because the

root of Kappa times beta because the beam splitter will change the amplitude

beam splitter will change the amplitude of the coherence from beta to square

of the coherence from beta to square root of Kappa beta so this will be the

root of Kappa beta so this will be the signal to noise ratio but if I inject

signal to noise ratio but if I inject squeeze vacuum into this port

squeeze vacuum into this port with enough squeezing these things uh

with enough squeezing these things uh this SNR goes to 4 beta squared as if

this SNR goes to 4 beta squared as if that loss never happened it's a very

that loss never happened it's a very very interesting observational squeeze

very interesting observational squeeze vacuum you're just injecting squeezing

vacuum you're just injecting squeezing into the quadrature that you're home

into the quadrature that you're home aligning

aligning um another thing

um another thing um that you can you know this was also

um that you can you know this was also people use this to propose multiple

people use this to propose multiple applications of this

applications of this squeezing inline squeezing so this is

squeezing inline squeezing so this is the same operation I showed in the

the same operation I showed in the previous slide where you go from vacuum

previous slide where you go from vacuum to squeezed vacuum so if I had say a

to squeezed vacuum so if I had say a coherent State beta

coherent State beta some loss transmissivity Kappa

some loss transmissivity Kappa and I have a homodyne detector here but

and I have a homodyne detector here but this time this homodyne detector has

this time this homodyne detector has some inefficiency meaning it is as if I

some inefficiency meaning it is as if I had a beam splitter of transmissivity

had a beam splitter of transmissivity data followed by an ideal homeowned

data followed by an ideal homeowned receiver

receiver if I were to put a squeezer with enough

if I were to put a squeezer with enough gain before that non-ideal

gain before that non-ideal inefficientness let's homodyne detection

inefficientness let's homodyne detection which has a sub Unity mixing efficiency

which has a sub Unity mixing efficiency detection efficiency product I can

detection efficiency product I can recover the effect of that lost photons

recover the effect of that lost photons before the homodyne by having enough

before the homodyne by having enough gain in that face sensitive amplifier or

gain in that face sensitive amplifier or a squeezer so these two effects of

a squeezer so these two effects of squeeze vacuum injection and phase

squeeze vacuum injection and phase sensitive amplification is something

sensitive amplification is something that we explored using this was many

that we explored using this was many years ago my my former PhD advisor Jeff

years ago my my former PhD advisor Jeff Shapiro and his colleague Prem Kumar and

Shapiro and his colleague Prem Kumar and and myself and a few others we were

and myself and a few others we were involved in this um DARPA program back

involved in this um DARPA program back in 2007.

in 2007. and uh in this we were looking at

and uh in this we were looking at um classical laser radar using a laser

um classical laser radar using a laser light to interrogate a scene

light to interrogate a scene um and uh we had a a pupil through which

um and uh we had a a pupil through which your light was coming back was a soft

your light was coming back was a soft aperture pupil so it was not a hard

aperture pupil so it was not a hard circular aperture it has a attenuation

circular aperture it has a attenuation and by injecting squeeze vacuum from

and by injecting squeeze vacuum from behind the aperture you could as if half

behind the aperture you could as if half the light coming from the scene see a

the light coming from the scene see a nice circular aperture as if there was

nice circular aperture as if there was no attenuation so that was the effect of

no attenuation so that was the effect of squeeze vacuum injection

squeeze vacuum injection and we had an imperfect hormone

and we had an imperfect hormone detection receiver which we preceded by

detection receiver which we preceded by a face sensitive amplifier before the

a face sensitive amplifier before the detector and we were able to get much

detector and we were able to get much better images of uh of saved the targets

better images of uh of saved the targets we were looking into this Homeward and

we were looking into this Homeward and detection radar so anyway so there are

detection radar so anyway so there are other many other applications of these

other many other applications of these Concepts that people have people have

Concepts that people have people have looked at

okay so let's now do some examples on Quantum Metrology with the tools we have

Quantum Metrology with the tools we have learned so we'll be using the tools that

learned so we'll be using the tools that I talked about and do some examples you

I talked about and do some examples you have seen some Heisenberg sensitivity

have seen some Heisenberg sensitivity examples in the previous lectures but

examples in the previous lectures but we'll now do some derivations

we'll now do some derivations this is a hike we did with kasturi when

this is a hike we did with kasturi when she visited my group for a couple of

she visited my group for a couple of um weeks a few months back and it was

um weeks a few months back and it was this beautiful waterfall then remember

this beautiful waterfall then remember going back to that same location in a

going back to that same location in a couple of months and there was no

couple of months and there was no waterfall and no water over here so

waterfall and no water over here so Tucson goes through these cycles of

Tucson goes through these cycles of rainfall

rainfall um and uh sometimes you can have

um and uh sometimes you can have completely different scenery when you

completely different scenery when you visit at different times

um so let's look at this canonical problem which Carl had stated in his

problem which Carl had stated in his lecture

lecture um the conjugate phase interferometer so

um the conjugate phase interferometer so I have Theta by 2 phase on one arm and

I have Theta by 2 phase on one arm and minus Theta by two in the other

minus Theta by two in the other and I will give you a photon budget in

and I will give you a photon budget in terms of an average mean or a mean

terms of an average mean or a mean photon number n that you can use so you

photon number n that you can use so you can use it to generate a coherent state

can use it to generate a coherent state or some other state or squeeze State

or some other state or squeeze State whatever you want

whatever you want and you want to estimate the phase

and you want to estimate the phase so with a coherent state I'm going to

so with a coherent state I'm going to use the coherence State let's say in

use the coherence State let's say in this way I'll split the coherent State

this way I'll split the coherent State on a 50 50 beam splitter

on a 50 50 beam splitter so if this Alpha were square root of 2

so if this Alpha were square root of 2 gets that phase Alpha over square root

gets that phase Alpha over square root of 2 that gets that phase

of 2 that gets that phase and now this is your Quantum state that

and now this is your Quantum state that encodes the phase

encodes the phase what would you do to find the Fischer

what would you do to find the Fischer information for that Quantum State

information for that Quantum State meaning at this point

meaning at this point I'm tasked with designing a receiver

I'm tasked with designing a receiver that has the highest sensitivity of

that has the highest sensitivity of estimating that parameter Theta so we've

estimating that parameter Theta so we've already learned the tools so what will I

already learned the tools so what will I do I'll calculate the the quantum

do I'll calculate the the quantum efficient information

efficient information and recall this formula that I wrote

and recall this formula that I wrote down when your Quantum state that

down when your Quantum state that carries Theta Theta is same as x x is

carries Theta Theta is same as x x is that unknown parameter so that's Theta

that unknown parameter so that's Theta here when rho of X is a pure State I can

here when rho of X is a pure State I can evaluate this this simple expression for

evaluate this this simple expression for the quantum facial information

the quantum facial information how would you go about doing this

how would you go about doing this calculation well

calculation well so as you see here you are going to need

so as you see here you are going to need a PSI dot right you have to

a PSI dot right you have to differentiate your coherent your your

differentiate your coherent your your state with respect to X

state with respect to X so what you would do is that if you

so what you would do is that if you remember that formula for a coherence

remember that formula for a coherence state

state um

um so coherent State I wrote down as was

so coherent State I wrote down as was that

that e to the minus 1 you should Alpha Square

e to the minus 1 you should Alpha Square over 2 Alpha to the N over root n

over 2 Alpha to the N over root n right so now instead here I have Alpha

right so now instead here I have Alpha over square root of 2

over square root of 2 so

so so that thing is 4

so that thing is 4 2 and through four

2 and through four and then this is Alpha over square root

and then this is Alpha over square root of 2.

of 2. and then there's a there's a phase

and then there's a there's a phase so that phase will come over here so

so that phase will come over here so it's e to the I

it's e to the I the N over 2. so take that expression so

the N over 2. so take that expression so this is not Alpha so Alpha over 2 e to

this is not Alpha so Alpha over 2 e to the I Theta over 2.

the I Theta over 2. and then uh you take the next cohere

and then uh you take the next cohere instead

instead so I wanted insert product so your Alpha

so I wanted insert product so your Alpha over square root of 2 e to the I Theta

over square root of 2 e to the I Theta over 2

over 2 um

um and another one is Alpha over square

and another one is Alpha over square root of 2 e to the minus I Theta over 2

root of 2 e to the minus I Theta over 2 . and for that you're gonna have another

. and for that you're gonna have another summation

and you will write the same thing to M the same coefficient but except that now

the same coefficient but except that now you'll have an e to the minus I Theta

you'll have an e to the minus I Theta over to end

over to end and that's your

and that's your that's your PSI of theta

that's your PSI of theta now you can take this expression take a

now you can take this expression take a derivative with respect to Theta so

derivative with respect to Theta so calculate PSI dot of theta

calculate PSI dot of theta and write down those inner products

and write down those inner products magnitude squared evaluate this

magnitude squared evaluate this expression leave it to you as a homework

expression leave it to you as a homework should try this out it's a very simple

should try this out it's a very simple calculation and what you should get is K

calculation and what you should get is K of theta equals to n for this setting

of theta equals to n for this setting so what what it means is that your there

so what what it means is that your there exists some measurement

exists some measurement such that the the variance of the

such that the the variance of the estimator that results from that

estimator that results from that measurement goes as one over

measurement goes as one over n times capital N where n is the number

n times capital N where n is the number of number of modes number of Trials of

of number of modes number of Trials of this measurement that you made with that

this measurement that you made with that same mean photon number Alpha coherent

same mean photon number Alpha coherent State prepared n times is that clear so

State prepared n times is that clear so this is I'm just Quantum official

this is I'm just Quantum official information remember it just tells you

information remember it just tells you that there exists some measurement that

that there exists some measurement that will do the job that will have that

will do the job that will have that sensitivity

sensitivity but now we are tasked with finding a

but now we are tasked with finding a measurement that achieves that

measurement that achieves that sensitivity so I told you that you can

sensitivity so I told you that you can calculate the sld and its eigenvectors

calculate the sld and its eigenvectors that's very hard for you to describe to

that's very hard for you to describe to an experimentalist like what does it

an experimentalist like what does it mean to calculate to do that experiment

mean to calculate to do that experiment the projective measurement defined by

the projective measurement defined by the sld eigenvectors but in this case

the sld eigenvectors but in this case it's actually not a hard to design

it's actually not a hard to design measurement so what we will do is that

measurement so what we will do is that we will just to recombine the two

we will just to recombine the two coherences another 50 50 beams later

coherences another 50 50 beams later so now you can do this math in your head

so now you can do this math in your head I don't need to write it down so this is

I don't need to write it down so this is Alpha

Alpha that coherent State here add these two

that coherent State here add these two and subtract these two what will happen

and subtract these two what will happen you will get a cosine and a sign right

you will get a cosine and a sign right you just take these two coefficients so

you just take these two coefficients so this plus this over square root of 2

this plus this over square root of 2 This plus this this minus this over

This plus this this minus this over square root of 2. now I have two

square root of 2. now I have two detectors

detectors they will both generate a poisson random

they will both generate a poisson random variable

variable whose mean is given by n cosine squared

whose mean is given by n cosine squared theta by 2 and N sine Square Theta by 2

theta by 2 and N sine Square Theta by 2 N is mod squared of alpha

N is mod squared of alpha okay so now how I need to calculate the

okay so now how I need to calculate the classical facial information of this

classical facial information of this measure because I have not specified the

measure because I have not specified the measurement you have a random variable

measurement you have a random variable so you have B of Y given X already and I

so you have B of Y given X already and I only need to calculate what is the

only need to calculate what is the official information so what's the first

official information so what's the first step well

step well take a look here we wrote down that the

take a look here we wrote down that the poisson distribution with an unknown

poisson distribution with an unknown mean the fission information is given by

mean the fission information is given by 1 by X right so the estimating Lambda 1

1 by X right so the estimating Lambda 1 from Z1 Lambda 1 being Let's uh it's a

from Z1 Lambda 1 being Let's uh it's a parameter which is the mean here this is

parameter which is the mean here this is official information

official information and then I also told you this Jacobian

and then I also told you this Jacobian rule

rule what if I don't care about Lambda 1 I

what if I don't care about Lambda 1 I actually care about Theta

actually care about Theta so I want to calculate the facial

so I want to calculate the facial information of theta it will be equal to

information of theta it will be equal to the fission information of Lambda 1

the fission information of Lambda 1 times the the squared of the derivative

times the the squared of the derivative of uh of this function f f one of theta

of uh of this function f f one of theta with respect to Theta squared

with respect to Theta squared so if you do this math again just a

so if you do this math again just a couple of lines you will get J1 equal to

couple of lines you will get J1 equal to n times sine Square Theta by 2.

n times sine Square Theta by 2. is that clear so I'm just calculating

is that clear so I'm just calculating the fission information for estimating

the fission information for estimating Theta from differential information for

Theta from differential information for estimating Lambda 1 using that Jacobian

estimating Lambda 1 using that Jacobian Rule now similarly the facial

Rule now similarly the facial information for estimating Theta from Z2

information for estimating Theta from Z2 is given by n times cosine squared Lam

is given by n times cosine squared Lam Theta by 2 you can calculate that using

Theta by 2 you can calculate that using the same method

the same method and because these two measurements are

and because these two measurements are statistically independent random

statistically independent random variables their Fischer informations add

variables their Fischer informations add and if you don't find the total official

and if you don't find the total official information for estimating Theta from

information for estimating Theta from both of these you can add these two fish

both of these you can add these two fish informations you get n again

informations you get n again which is really cool which means that

which is really cool which means that now if I were to write down the minimum

now if I were to write down the minimum maximum likelihood estimator on Z1 and

maximum likelihood estimator on Z1 and Z2 my variance of the estimator is going

Z2 my variance of the estimator is going to go as 1 over n n which is exactly

to go as 1 over n n which is exactly what the quantum pressure information

what the quantum pressure information did

did which means in this particular case is

which means in this particular case is that this measurement is a Quantum

that this measurement is a Quantum optimal measurement is this the

optimal measurement is this the projection onto the sld eigenvectors it

projection onto the sld eigenvectors it is not

is not okay this is a different way I can write

okay this is a different way I can write down the quantum description of this

down the quantum description of this measurement

measurement how I can take the fog basis go

how I can take the fog basis go backwards so this beam splitter it will

backwards so this beam splitter it will be a crazy looking measurement basis

be a crazy looking measurement basis it's not the sld eigen measurement basis

it's not the sld eigen measurement basis necessarily so but it achieves the qfi

necessarily so but it achieves the qfi nevertheless

nevertheless all right let's take another example of

all right let's take another example of something called a noon state in the

something called a noon state in the literature it's a very very well known

literature it's a very very well known state in the context of quantum

state in the context of quantum Metrology in fact one of the earliest

Metrology in fact one of the earliest examples of quantum limited sensitivity

examples of quantum limited sensitivity for estimation phase estimation that

for estimation phase estimation that came what described as with respect to a

came what described as with respect to a noon state so noon state but but I will

noon state so noon state but but I will pretend n is an integer

pretend n is an integer um

um and that there's a fox state with photon

and that there's a fox state with photon number n so this has a mean photon

number n so this has a mean photon number of n right

number of n right um and then I will have the state of the

um and then I will have the state of the output and remember I said that if a

output and remember I said that if a phase acts on a fox state it gets this

phase acts on a fox state it gets this uh phase in front of the fox state but

uh phase in front of the fox state but now I'll get two different phases e to

now I'll get two different phases e to the I Theta by 2 because the fox State

the I Theta by 2 because the fox State hits this mode uh and this foxt state

hits this mode uh and this foxt state hits that mode here now you can do the

hits that mode here now you can do the same thing go ahead and calculate the

same thing go ahead and calculate the the quantum efficient information in

the quantum efficient information in fact you will find this calculation a

fact you will find this calculation a lot easier than the coherent State One

lot easier than the coherent State One so the side dot you are just going to

so the side dot you are just going to take this see you will differentiate

take this see you will differentiate with respect to psi you will get a

with respect to psi you will get a i n over 2 e to the i n Theta over 2

i n over 2 e to the i n Theta over 2 will get a minus i n over 2 to the i n

will get a minus i n over 2 to the i n Theta over to do the differentiation do

Theta over to do the differentiation do these inner products and if you do the

these inner products and if you do the math you should see K of theta will come

math you should see K of theta will come out as N squared

out as N squared what does it mean well it means that

what does it mean well it means that there exists a measurement on this

there exists a measurement on this output such that the variance of the

output such that the variance of the estimator will go as one over n times N

estimator will go as one over n times N squared so little n is still a number of

squared so little n is still a number of copies of the state

copies of the state so this is what is you know called the

so this is what is you know called the Heisenberg limited sensitivity

Heisenberg limited sensitivity um in this case I will not show an

um in this case I will not show an actual example calculation of a receiver

actual example calculation of a receiver even though that same interferometric

even though that same interferometric receiver for the coherent State actually

receiver for the coherent State actually works in this case also but I just

works in this case also but I just wanted to show you something little

wanted to show you something little different

different I will Define a measurement by these two

I will Define a measurement by these two projectors

projectors where these two projectors oh I didn't

where these two projectors oh I didn't even write down these projectors so PSI

even write down these projectors so PSI one I intended it to be

i1 is n 0 plus 0 n over root 2

over root 2 and PSI 2 is n 0 minus 0 n over root 2.

now these two are mutually orthogonal States and hence describe a valid point

States and hence describe a valid point of measurement

of measurement and then if you were to work out the

and then if you were to work out the probability of getting the one outcome

probability of getting the one outcome or the two outcome you will just take

or the two outcome you will just take the magnitude Square inner product with

the magnitude Square inner product with respect to the noon state

respect to the noon state and you get these two formulas cosine

and you get these two formulas cosine squared and sine squared and N Theta by

squared and sine squared and N Theta by 2.

2. what do we have here so we have exactly

what do we have here so we have exactly that setting that we derived earlier

that setting that we derived earlier you have little n copies of a Bernoulli

you have little n copies of a Bernoulli random variable

random variable with a parameter P which is the

with a parameter P which is the parameter that you care about

parameter that you care about but not quite you don't care about P you

but not quite you don't care about P you actually care about Theta so you have to

actually care about Theta so you have to still use that Jacobian

still use that Jacobian right

right so first question is that how do I

so first question is that how do I calculate the classical facial

calculate the classical facial information of this measurement well the

information of this measurement well the first thing is that recall from that

first thing is that recall from that formula over here on the board that the

formula over here on the board that the official information for calculating for

official information for calculating for estimating p is 1 over P times 1 minus P

estimating p is 1 over P times 1 minus P we just derive that for the Bernoulli

we just derive that for the Bernoulli random variable

random variable but then if you want to estimate Theta

but then if you want to estimate Theta so the J of theta is given by J 1 of

so the J of theta is given by J 1 of this J of P times this this mod squared

this J of P times this this mod squared the Jacobian F Prime of theta when F

the Jacobian F Prime of theta when F Prime of theta is just the derivative of

Prime of theta is just the derivative of the functional form of theta be written

the functional form of theta be written as F of theta take this differentiate

as F of theta take this differentiate that square that multiply by this thing

that square that multiply by this thing do a couple of lines of math you will

do a couple of lines of math you will see G of theta equal to N squared

see G of theta equal to N squared which means that this particular

which means that this particular projective measurement achieves the

projective measurement achieves the quantum fission information

quantum fission information um and uh again it shows that the

um and uh again it shows that the classical facial information of a

classical facial information of a measurement again just like the previous

measurement again just like the previous example equals the quantification

example equals the quantification information which means that this

information which means that this measurement is Optimum or one of the

measurement is Optimum or one of the optimal measurements and hence the

optimal measurements and hence the maximum likelihood estimator at the

maximum likelihood estimator at the output will achieve that same scaling

output will achieve that same scaling all right so let's look at using a

all right so let's look at using a squeeze light probe with the same mean

squeeze light probe with the same mean photon number in this case I will not

photon number in this case I will not have the mathematical tools to show you

have the mathematical tools to show you the full calculation but this is the

the full calculation but this is the example which I want to use for some

example which I want to use for some photonic sensing applications that I

photonic sensing applications that I will talk about afterwards

will talk about afterwards okay so now if I take a squeezed vacuum

okay so now if I take a squeezed vacuum state with uh with the squeezing

state with uh with the squeezing parameter squeeze this way

parameter squeeze this way and I squeezed display squeeze state

and I squeezed display squeeze state with a mean of Alpha and a squeezing

with a mean of Alpha and a squeezing parameter of PSI the same squeezing

parameter of PSI the same squeezing parameter

parameter I'll get a state here I calculate the

I'll get a state here I calculate the official information this also goes as N

official information this also goes as N squared okay and in this case the

squared okay and in this case the measurement that works to achieve this

measurement that works to achieve this is a homonym detection measurement on

is a homonym detection measurement on only one of the quadratures after you

only one of the quadratures after you put it through a 50 50 games later so if

put it through a 50 50 games later so if you do a homodide measurement write down

you do a homodide measurement write down an estimator if you calculate the

an estimator if you calculate the variance of this estimator you will get

variance of this estimator you will get the variance of the estimator to be

the variance of the estimator to be equal as 1 over N squared and here we

equal as 1 over N squared and here we arrange the squeezing such that the

arrange the squeezing such that the total mean photon number of these two

total mean photon number of these two squeeze States is equal to n capital N

squeeze States is equal to n capital N just to be fair in my calculation

just to be fair in my calculation all right so this this was used by my

all right so this this was used by my former student Michael in designing a

former student Michael in designing a Quantum enhanced fiber optical gyroscope

Quantum enhanced fiber optical gyroscope so gyroscopes are a device that can be

so gyroscopes are a device that can be used to measure you know acceleration

used to measure you know acceleration for example so yeah in a normal

for example so yeah in a normal gyroscope if a laser gyroscope you have

gyroscope if a laser gyroscope you have a laser going into a long Loop and into

a laser going into a long Loop and into homodyne detection

homodyne detection and in this case he injected squeezed

and in this case he injected squeezed vacuum through a circulator and then

vacuum through a circulator and then looked at some applications of that not

looked at some applications of that not just with a single mode squeeze vacuum

just with a single mode squeeze vacuum but extensions of that where you split

but extensions of that where you split the squeeze back into multiple modes to

the squeeze back into multiple modes to get an entangle State and get further

get an entangle State and get further better performance but again I will not

better performance but again I will not go into the details of this calculation

go into the details of this calculation here

here now the moment you have loss in any one

now the moment you have loss in any one of these sensing examples what happens

of these sensing examples what happens is that this Heisenberg limited

is that this Heisenberg limited sensitivity goes away so see the

sensitivity goes away so see the variance of the noon State or the

variance of the noon State or the squeeze State transmitter that goes 1

squeeze State transmitter that goes 1 over N squared the variance for the

over N squared the variance for the coherency transmitter goes as 1 over n

coherency transmitter goes as 1 over n but if you have loss then what happens

but if you have loss then what happens is that it starts scaling like the

is that it starts scaling like the quantum case for low mean photon number

quantum case for low mean photon number and when you go to High n it becomes

and when you go to High n it becomes parallel to 1 by n

parallel to 1 by n so basically there is no Heisenberg

so basically there is no Heisenberg limited sensitivity as a scaling law for

limited sensitivity as a scaling law for large n you always go back to 1 over n

large n you always go back to 1 over n scaling for the variance but you can

scaling for the variance but you can still get a good constant Factor

still get a good constant Factor Improvement in the variance using

Improvement in the variance using Quantum Resources and how much

Quantum Resources and how much improvement you get depends upon how

improvement you get depends upon how lost your system is

lost your system is okay so the last example I want to do is

okay so the last example I want to do is the magnetic field sensing with two

the magnetic field sensing with two level atoms

level atoms and in this example it's a very very

and in this example it's a very very simple example you have a state of a

simple example you have a state of a qubit zero plus one over two root 2 and

qubit zero plus one over two root 2 and as a function of time you get um

as a function of time you get um uh you get a Time varying a Time varying

uh you get a Time varying a Time varying phase being applied to it and what you

phase being applied to it and what you want to estimate is the is Theta meaning

want to estimate is the is Theta meaning the rate at which this phase is

the rate at which this phase is oscillating okay so this Theta is what

oscillating okay so this Theta is what you care about

you care about and what you can do is that if you take

and what you can do is that if you take the state and make a measurement in this

the state and make a measurement in this 45 degree rotated basis you will get a

45 degree rotated basis you will get a Bernoulli outcome again P or 1 minus P

Bernoulli outcome again P or 1 minus P this these two probabilities will be the

this these two probabilities will be the plus outcome or the minus outcome it's

plus outcome or the minus outcome it's very easy to calculate from this right

very easy to calculate from this right how would you calculate from this you

how would you calculate from this you will take a plus inner product with this

will take a plus inner product with this I of T magnitude squared you will get

I of T magnitude squared you will get this formula minus the state inner

this formula minus the state inner product with PSI magnitude Square you

product with PSI magnitude Square you will get sine squared theta T by 2. so

will get sine squared theta T by 2. so we know that the fission information is

we know that the fission information is given by that formula again 1 over P

given by that formula again 1 over P times 1 minus p

times 1 minus p and apply the Jacobian again your J of

and apply the Jacobian again your J of theta will come to t squared

theta will come to t squared so what it means is that the variance of

so what it means is that the variance of your ml estimator will go as one over

your ml estimator will go as one over little n times t squared with little n

little n times t squared with little n here is the number of atoms that is

here is the number of atoms that is let's say sensing the same magnetic

let's say sensing the same magnetic field so each atom is being subject to

field so each atom is being subject to the same field

the same field and you want to write down the variance

and you want to write down the variance of the estimator for estimating that

of the estimator for estimating that that that parameter Theta which is the

that that parameter Theta which is the rate of oscillation

rate of oscillation now what if you had these n atoms that

now what if you had these n atoms that were

were initialize instead of all of them in the

initialize instead of all of them in the equal superposition of one and zero in

equal superposition of one and zero in this g8z state or what Carl called the

this g8z state or what Carl called the cat state zero zero zero zero zero plus

cat state zero zero zero zero zero plus one one one one one you will get that

one one one one one you will get that phase e to the i n times Theta times T

phase e to the i n times Theta times T in front of that term

in front of that term and again you can do a measurement on

and again you can do a measurement on this in The Logical plus minus basis

this in The Logical plus minus basis meaning these two vectors

meaning these two vectors and you will get again a Bernoulli

and you will get again a Bernoulli outcome but with a probability cosine

outcome but with a probability cosine squared theta and T over 2 and 1 minus q

squared theta and T over 2 and 1 minus q and again I'm losing the same formula my

and again I'm losing the same formula my J of theta will simply come to be NT

J of theta will simply come to be NT Bank total square it is like I am doing

Bank total square it is like I am doing the same calculation as before like J if

the same calculation as before like J if you look at this here

you look at this here uh this J of theta is t squared

uh this J of theta is t squared where T was this term over here

where T was this term over here um and uh and the T appeared in the in

um and uh and the T appeared in the in the Bernoulli probably distribution is

the Bernoulli probably distribution is cosine squared theta T by 2. so if I

cosine squared theta T by 2. so if I replace T by n t

replace T by n t the J the whole calculation is the same

the J the whole calculation is the same this becomes since the p square becomes

this becomes since the p square becomes NT times whole squared

NT times whole squared and because I just had one copy of this

and because I just had one copy of this state

state I don't need a little n multiplying

I don't need a little n multiplying outside my ml estimator will be just 1 1

outside my ml estimator will be just 1 1 over J of theta so it's n Square t

over J of theta so it's n Square t squared

squared so what you see here is that the main

so what you see here is that the main difference between the previous slide is

difference between the previous slide is that P Square is the same but that n has

that P Square is the same but that n has become N squared now so this is the

become N squared now so this is the Heisenberg limited sensitivity of field

Heisenberg limited sensitivity of field field sensing

field sensing okay so let's see I am now at 3 30.

okay so let's see I am now at 3 30. um

um and I was I'm going to talk about

and I was I'm going to talk about network of sensors and how do I create

network of sensors and how do I create these Atomic sensor networks and a

these Atomic sensor networks and a couple of examples of that so what do

couple of examples of that so what do you think should I stop here or go for

you think should I stop here or go for another 15 minutes this section

another 15 minutes this section um and uh what's your what's your

um and uh what's your what's your preference

but yeah that's that sounds good that sounds

yeah that's that sounds good that sounds good yeah that's perfectly fine so if

good yeah that's perfectly fine so if you want to take questions at this point

you want to take questions at this point so my plan for the next part here is to

so my plan for the next part here is to talk about uh not just a single sensor

talk about uh not just a single sensor if you have multiple sensors working

if you have multiple sensors working together towards one task

together towards one task um then uh how that how can entanglement

um then uh how that how can entanglement among Those sensors gives you a better

among Those sensors gives you a better sensitivity and then I'll show examples

sensitivity and then I'll show examples of such sensor networks for magnetic

of such sensor networks for magnetic field sensing for RF photonics sensors

field sensing for RF photonics sensors for things like long Baseline astronomy

for things like long Baseline astronomy and one example in multiple spatial

and one example in multiple spatial modes that are entangled for higher

modes that are entangled for higher Precision beam deflection measurements

Precision beam deflection measurements so my talk is going to get more and more

so my talk is going to get more and more and more applied as I go towards the

and more applied as I go towards the latter parts of my talk

latter parts of my talk but any questions yeah so

but any questions yeah so questions

questions sure

oh uh sir I have a question with the with regards to the estimator section uh

with regards to the estimator section uh you talked about uh how the facial

you talked about uh how the facial information may or may not depend on the

information may or may not depend on the variable so uh is that uh dependent only

variable so uh is that uh dependent only on the type of distribution you have or

on the type of distribution you have or is there anything else that may uh

is there anything else that may uh there's nothing other than the type of

there's nothing other than the type of the distribution

the distribution um in general the facial information

um in general the facial information will will be a function of the parameter

will will be a function of the parameter when it is not is what I consider as the

when it is not is what I consider as the special case like that gaussian

special case like that gaussian distribution with an unknown mean and a

distribution with an unknown mean and a known variance definition is one over

known variance definition is one over the variance squared but I would say

the variance squared but I would say that is more of a

that is more of a exception than the rule typical

exception than the rule typical efficient information is a function of

efficient information is a function of the parameter

the parameter you will see one more example in my talk

you will see one more example in my talk tomorrow

tomorrow on Quantum limited

on Quantum limited estimation of the separation between two

estimation of the separation between two stars they are also you'll see facial

stars they are also you'll see facial information does not depend on the

information does not depend on the parameter you care about but that's uh

parameter you care about but that's uh mostly it does mostly it does yeah

mostly it does mostly it does yeah and also the basic question regarding

and also the basic question regarding the squeezing of light so you talked

the squeezing of light so you talked about the noise ball right so I'm how I

about the noise ball right so I'm how I understand is that when you squeeze the

understand is that when you squeeze the light in towards one quadrature so the

light in towards one quadrature so the uncertainty increases in the other

uncertainty increases in the other orthogonal quadrature so the noise wall

orthogonal quadrature so the noise wall is is that the uncertainty is is the

is is that the uncertainty is is the uncertainty that noise ball is it's like

uncertainty that noise ball is it's like that natagram I had the coherent State

that natagram I had the coherent State oscillatory component and the noise

oscillatory component and the noise balls vertical length was the same

balls vertical length was the same which simply means that if I did the

which simply means that if I did the homodyne detection no matter which

homodyne detection no matter which quadrature of the complex number I am

quadrature of the complex number I am trying to measure real part imaginary

trying to measure real part imaginary but I'll still always get one fourth

but I'll still always get one fourth that's the variance in the photon number

that's the variance in the photon number units for a squeeze State as I'm going

units for a squeeze State as I'm going through that 0 to 2 pi oscillation some

through that 0 to 2 pi oscillation some part of the zero to two Pi oscillation I

part of the zero to two Pi oscillation I increase it it means that the other

increase it it means that the other orthogonal quantity drama you squeeze it

orthogonal quantity drama you squeeze it so it goes like this

okay so uh you have talked about uh uh uh uh exquisition said has even number

uh uh exquisition said has even number of photons and mathematically so can you

of photons and mathematically so can you give a physical intuition why uh we get

give a physical intuition why uh we get only even number of photons physical

only even number of photons physical intuition

intuition you have an answer Psy to that I what is

you have an answer Psy to that I what is the physical intuition for the squeeze

the physical intuition for the squeeze vacuum squeeze around the real

vacuum squeeze around the real quadrature it would have only even

quadrature it would have only even number of protons

it is quadrature dependent I'm trying so his question is a physical reason for

his question is a physical reason for why uh okay I can give you

why uh okay I can give you not for squeeze vacuum but you know

not for squeeze vacuum but you know there is another state that I did not

there is another state that I did not talk about it's called the uh have you

talk about it's called the uh have you heard of the term cat State yes so if I

heard of the term cat State yes so if I take a coherence State Alpha that I

take a coherence State Alpha that I wrote down earlier so e to the minus mod

wrote down earlier so e to the minus mod Alpha Square over 2 Alpha to the N

Alpha Square over 2 Alpha to the N square root of n factorial n

square root of n factorial n if I Define a state Alpha plus minus

if I Define a state Alpha plus minus Alpha

Alpha with whatever normalization constant I

with whatever normalization constant I need in order to make it a unit Norm

need in order to make it a unit Norm state

state what will happen if I write the state

what will happen if I write the state can you see what happens here

can you see what happens here when I take minus Alpha

when I take minus Alpha I have the first part of this sum

I have the first part of this sum Remains the Same e to the minus mod

Remains the Same e to the minus mod Alpha Square by 2 but here I get Alpha

Alpha Square by 2 but here I get Alpha to the N plus minus Alpha to the n

this state is obviously not a Pokemon State this is not poisson but if you see

State this is not poisson but if you see these terms

these terms they are um

they are um uh they are they go between zero and

uh they are they go between zero and non-zero for odd and even right so when

non-zero for odd and even right so when so this term is minus 1 to the N times

so this term is minus 1 to the N times Alpha to the N so when n equal to 1 this

Alpha to the N so when n equal to 1 this is negative so it cancels out so it only

is negative so it cancels out so it only has even components but now this I did

has even components but now this I did not give you an answer for a squeeze

not give you an answer for a squeeze vacuum but if you look at the wigner

vacuum but if you look at the wigner function of the

function of the this particular cat state is a very

this particular cat state is a very strong resemblance with the squeeze

strong resemblance with the squeeze vacuum squeeze along that that q

vacuum squeeze along that that q quadrature that I had drawn but I don't

quadrature that I had drawn but I don't know the short answer is I don't have a

know the short answer is I don't have a very good answer to the physical

very good answer to the physical intuition

intuition or why you have

or why you have uh one other way to think about

uh one other way to think about um you know squeezing the first paper

um you know squeezing the first paper that introduced squeezed light

that introduced squeezed light um was a three-part paper in 1979 to

um was a three-part paper in 1979 to 1981 by Jeff Shapiro and Horace un

1981 by Jeff Shapiro and Horace un they did not call it school slide at the

they did not call it school slide at the pipe so if you look at the title of the

pipe so if you look at the title of the paper they called it two Photon coherent

paper they called it two Photon coherent States

States now their reason for calling it a two

now their reason for calling it a two Photon coherence state is that the

Photon coherence state is that the squeezing hamiltonian

squeezing hamiltonian if you write down the unitary of

if you write down the unitary of squeezing operator

squeezing operator it looks like this

these are the field operators and this is a annihilation of photons

and this is a annihilation of photons twice in the structure for Photon twice

twice in the structure for Photon twice so that was the reason why they called

so that was the reason why they called it a two Photon

it a two Photon um two Photon coherent State and it was

um two Photon coherent State and it was Carl caves who started calling it

Carl caves who started calling it squeeze State because of its picture in

squeeze State because of its picture in the Victor space but that is another way

the Victor space but that is another way to think about why the squeeze state has

to think about why the squeeze state has even number of photons if you were to

even number of photons if you were to work out the fog basis elements because

work out the fog basis elements because a squeezed vacuum will be simply

a squeezed vacuum will be simply applying this operator to vacuum

okay and if you work out the application so now if I have an e to the N operator

so now if I have an e to the N operator I will have to write it as you know the

I will have to write it as you know the usual Taylor series

usual Taylor series a square over 2 factorial and so on

a square over 2 factorial and so on take this

take this do this thing and apply it to vacuum and

do this thing and apply it to vacuum and you will be able to see that you will

you will be able to see that you will get 0 and then 2 and 4 and 6 and so on

get 0 and then 2 and 4 and 6 and so on the middle things will cancel out this

the middle things will cancel out this is the closest I can come in terms of

is the closest I can come in terms of physical intuition

physical intuition hope that's um

hope that's um thank you

thank you three four five okay uh yeah so you

three four five okay uh yeah so you mentioned about svi and PSA techniques

mentioned about svi and PSA techniques so I was wondering if there is a

so I was wondering if there is a threshold for the squeezing parameter

threshold for the squeezing parameter that you actually need to uh see uh the

that you actually need to uh see uh the independency from copper and if it's a

independency from copper and if it's a good question there is no threshold it's

good question there is no threshold it's a smooth thing right so for Kappa to

a smooth thing right so for Kappa to completely disappear your R will have to

completely disappear your R will have to be infinity or your ETA for the homody

be infinity or your ETA for the homody and fish inefficiency to completely go

and fish inefficiency to completely go to one your face sensitive amplifiers

to one your face sensitive amplifiers gain has to go to Infinity

gain has to go to Infinity but obviously that is not the case in an

but obviously that is not the case in an experiment so in that experiment we had

experiment so in that experiment we had six DBS squeezing for the squeeze vacuum

six DBS squeezing for the squeeze vacuum and something like 2 DB of squeezing for

and something like 2 DB of squeezing for the PSA so we just reduced the the the

the PSA so we just reduced the the the effect of that inefficiency in the

effect of that inefficiency in the homonym receiver and hence improve the

homonym receiver and hence improve the quality of our image in the experiment

quality of our image in the experiment so it is as if I have a homodyne

so it is as if I have a homodyne detector with a detection efficient 70

detector with a detection efficient 70 but as if my information bearing light

but as if my information bearing light is seeing a homodyne detector of 80

is seeing a homodyne detector of 80 efficiency

efficiency so that was the effect that we had seen

so that was the effect that we had seen But there is no sharp threshold it's a

But there is no sharp threshold it's a smooth effect okay

hello sir so for the current state of light the it follows the poisonian

light the it follows the poisonian statistics okay so it has a probability

statistics okay so it has a probability distribution function and the squeeze

distribution function and the squeeze state of light it follows the supposed

state of light it follows the supposed Union statistics so where the variance

Union statistics so where the variance is less than the mean so does it have

is less than the mean so does it have any analytical distribution function

any analytical distribution function just like the current state oh yeah of

just like the current state oh yeah of course of course the sun slides Fox fog

course of course the sun slides Fox fog distribution I don't like writing that

distribution I don't like writing that it's a not a very certain distribution

it's a not a very certain distribution to write down but you can find it in

to write down but you can find it in textbooks it involves the harmite

textbooks it involves the harmite polynomials but definitely there is a

polynomials but definitely there is a nice close form expression for that

nice close form expression for that distribution so the thermal State also

distribution so the thermal State also has that kind of distribution so the

has that kind of distribution so the thermal States Photon distribution is a

thermal States Photon distribution is a what is called the Bose Einstein

what is called the Bose Einstein distribution okay so that is or in the

distribution okay so that is or in the language of Statistics it's a geometric

language of Statistics it's a geometric distribution

distribution thermal States Photon distribution is a geometric distribution

distribution is a geometric distribution squeeze test number distribution is a

squeeze test number distribution is a much more complicated looking

much more complicated looking distribution in termite polynomial

distribution in termite polynomial so so if we converge the current state

so so if we converge the current state and the thermostats do we get the uh

and the thermostats do we get the uh suppose onion so coherence State and

suppose onion so coherence State and thermal State they are both classical

thermal State they are both classical States so if you mix them let's say on a

States so if you mix them let's say on a beam splitter what you will get is a

beam splitter what you will get is a displaced thermal state in fact on the

displaced thermal state in fact on the two outputs of the beam splitter you

two outputs of the beam splitter you will get a classically correlated

will get a classically correlated gaussian state with a non-zero mean it

gaussian state with a non-zero mean it will not be a squeeze State a squeeze

will not be a squeeze State a squeeze state is a non-classical state

state is a non-classical state so if I put a squeeze State through a

so if I put a squeeze State through a beam splitter you will get an entangle

beam splitter you will get an entangle set on the two outputs and that's what

set on the two outputs and that's what I'm going to go into the next part of my

I'm going to go into the next part of my talk on Quantum sensor networks but with

talk on Quantum sensor networks but with thermal State and a coherent State you

thermal State and a coherent State you can all kinds of things that you can

can all kinds of things that you can generate with data and linear Optics

generate with data and linear Optics beam Splitters are states that are only

beam Splitters are states that are only classically correlated not entangled

classically correlated not entangled okay

okay I think I saw a question there for the

I think I saw a question there for the measurement you have chosen the specific

measurement you have chosen the specific projectors what is the reason behind

projectors what is the reason behind that you have chosen specific projectors

that you have chosen specific projectors and that is also in entangled basis that

and that is also in entangled basis that PSI 1 and PSI 2 so the for the noon

PSI 1 and PSI 2 so the for the noon State example I showed yes yes I just

State example I showed yes yes I just picked just I just picked one

picked just I just picked one measurement basis just to show that that

measurement basis just to show that that measurements CFI equals the qfi just to

measurements CFI equals the qfi just to show you an example

show you an example like

like when you are finding optimal receivers

when you are finding optimal receivers often times for many problems in

often times for many problems in communication sensing I work with it's

communication sensing I work with it's it's actually

it's actually R art slash intuition finding optimal

R art slash intuition finding optimal receivers Quantum tools they give you a

receivers Quantum tools they give you a good measurable value this is the best

good measurable value this is the best you can do from there finding how do you

you can do from there finding how do you design a measurement that achieves that

design a measurement that achieves that there is no one recipe so I just wanted

there is no one recipe so I just wanted to show one example calculation for the

to show one example calculation for the coherent State one it was very obvious

coherent State one it was very obvious as to why we should have used that means

as to why we should have used that means better

better for the newstead example that

for the newstead example that measurement was the obvious because noon

measurement was the obvious because noon or not known meaning the up the the

or not known meaning the up the the orthogonal state n 0 minus 0 n it was

orthogonal state n 0 minus 0 n it was the obvious one to uh to get that five

the obvious one to uh to get that five because that Phi appears in a Bernoulli

because that Phi appears in a Bernoulli distribution probability which is

distribution probability which is something that is easy to estimate but

something that is easy to estimate but other than that there's no real

other than that there's no real intuition behind it

intuition behind it thanks

thanks but an actual measurement that will

but an actual measurement that will achieve it actually a beam splitter

achieve it actually a beam splitter followed by Photon detection would have

followed by Photon detection would have achieved it I just did not want to do

achieved it I just did not want to do the calculation because interfering Fox

the calculation because interfering Fox States through a beam spirit is much

States through a beam spirit is much more difficult to do in Us in this

more difficult to do in Us in this setting but just wanted to say that

setting but just wanted to say that there are many many uh measurements for

there are many many uh measurements for the same problem that will achieve the

the same problem that will achieve the qfi

qfi which are not even sometimes the sld

which are not even sometimes the sld measurement

measurement uh sir my question is regarding the

uh sir my question is regarding the experimental schematics that you've

experimental schematics that you've shown uh where the you've placed two

shown uh where the you've placed two phase differences after the beam

phase differences after the beam splitter uh five by two and minus five

splitter uh five by two and minus five by two is it always necessary to have

by two is it always necessary to have five by two and minus five by two no uh

five by two and minus five by two no uh will that was just one example the five

will that was just one example the five by two and minus five by two example

by two and minus five by two example this is uh people who call it a

this is uh people who call it a conjugate phase interferometer yeah

conjugate phase interferometer yeah conjugate phase interferometer comes

conjugate phase interferometer comes about in a sanyac loop-based fiber

about in a sanyac loop-based fiber optical gyroscope okay okay but if you

optical gyroscope okay okay but if you have a phase only on one arm that also

have a phase only on one arm that also is fine like the ligo example that Carl

is fine like the ligo example that Carl talked about but will the calculations

talked about but will the calculations change significantly not significantly

change significantly not significantly not see no you can in fact I you can try

not see no you can in fact I you can try the calculation with just Theta on one

the calculation with just Theta on one arm with a coherent State and I think

arm with a coherent State and I think your official information for that

your official information for that instead of n will come out to be 4 n if

instead of n will come out to be 4 n if I remember correctly okay but try it out

I remember correctly okay but try it out it's a it will go as always proportional

it's a it will go as always proportional to n okay and with the squeeze state or

to n okay and with the squeeze state or noon state it will go as the constant

noon state it will go as the constant will change depending upon your exact

will change depending upon your exact setting uh and so one more question so

setting uh and so one more question so uh you've shown the fission

uh you've shown the fission in terms of uh the derivative of PSI and

in terms of uh the derivative of PSI and also PSI I mean it is in in the form

also PSI I mean it is in in the form having these two things some places have

having these two things some places have also seen it being represented as the

also seen it being represented as the variance of the hamiltonian with the

variance of the hamiltonian with the Hamilton so how are these two related

Hamilton so how are these two related how are they related very good question

how are they related very good question you must have noticed in my talk I

you must have noticed in my talk I didn't have a hamiltonian yeah right

didn't have a hamiltonian yeah right yeah I was directly asked writing rho of

yeah I was directly asked writing rho of theta rho of X yeah where did row of X

theta rho of X yeah where did row of X come from in a real sensing setting uh

come from in a real sensing setting uh well I should not say real sensing the

well I should not say real sensing the two types of sensors one is passive

two types of sensors one is passive sensors

sensors one is active sensors like when I take

one is active sensors like when I take this phone and take a camera a picture

this phone and take a camera a picture this is a passive sensor I'm not sending

this is a passive sensor I'm not sending any light yeah but I can still write

any light yeah but I can still write down the quantum state of the photons

down the quantum state of the photons that I'm collecting I'll show some

that I'm collecting I'll show some examples tomorrow of such passive

examples tomorrow of such passive sensors there

sensors there you will oftentimes just write the

you will oftentimes just write the quantum State based on your physics of

quantum State based on your physics of the Imaging system

the Imaging system but then if I'm sending a laser light

but then if I'm sending a laser light probe or a squeeze light probe for a

probe or a squeeze light probe for a beam reflection measurement or uh for

beam reflection measurement or uh for the ligo interferometer there I can

the ligo interferometer there I can describe the action of that measurement

describe the action of that measurement of that physical

of that physical um Metrology tool to my probe and that

um Metrology tool to my probe and that action physicists would often write as a

action physicists would often write as a hamiltonian which is e to the i h t uh

hamiltonian which is e to the i h t uh where T is the time duration over which

where T is the time duration over which you're acting upon the state with that

you're acting upon the state with that hamiltonian that's a unitary now e to

hamiltonian that's a unitary now e to the i h t that H will depend on that

the i h t that H will depend on that Theta right yeah and e to the iht

Theta right yeah and e to the iht applied on the state size 0 will give

applied on the state size 0 will give you your final state which you can then

you your final state which you can then think as my PSI of theta or rho of theta

think as my PSI of theta or rho of theta you calculate the variance of the

you calculate the variance of the hamiltonian it will match the formula I

hamiltonian it will match the formula I gave you if you were to write that on

gave you if you were to write that on PSI of theta okay okay got it okay and

PSI of theta okay okay got it okay and just one last question so uh in the part

just one last question so uh in the part that you showed that you were

that you showed that you were experimentally generating The Squeeze

experimentally generating The Squeeze state so you said that you actually

state so you said that you actually squeezed the vacuum so basically

squeezed the vacuum so basically according to me I mean you're not

according to me I mean you're not sending anything in so I mean through

sending anything in so I mean through the crystal so what exactly what exactly

the crystal so what exactly what exactly am I doing yeah okay very good question

am I doing yeah okay very good question so that's the whole that could be a

so that's the whole that could be a subject matter for a whole uh two

subject matter for a whole uh two lecture series

lecture series um

um non-linear Optics okay classical

non-linear Optics okay classical non-linear Optics there are many

non-linear Optics there are many processes like down conversion some

processes like down conversion some frequency generation difference

frequency generation difference frequency generation

frequency generation so the process that we are using in my

so the process that we are using in my lab is called the spontaneous parametric

lab is called the spontaneous parametric down conversion so what happens there is

down conversion so what happens there is that there are three frequencies

that there are three frequencies involved so let's say there is a pump

involved so let's say there is a pump frequency fifth so pump frequency

frequency fifth so pump frequency if you and there are two more

if you and there are two more frequencies I call the signal frequency

frequencies I call the signal frequency in idler frequency

in idler frequency so I put nothing in the signal and idler

so I put nothing in the signal and idler I put vacuum in signal and idler okay

I put vacuum in signal and idler okay pump pump I put a strong coherence state

pump pump I put a strong coherence state now if you do the classical non-linear

now if you do the classical non-linear Optics of the input output Theory then

Optics of the input output Theory then solve the Maxwell's equations in a bulk

solve the Maxwell's equations in a bulk Chi 2 medium

Chi 2 medium what you will get is that you will get a

what you will get is that you will get a coupled mode equation in terms of the

coupled mode equation in terms of the amplitudes of all these three modes of

amplitudes of all these three modes of the three okay three frequency modes if

the three okay three frequency modes if you look at those couple more equations

you look at those couple more equations you will see that

you will see that if a s and a i the complex numbers

if a s and a i the complex numbers corresponding to the signal and idler

corresponding to the signal and idler frequencies were zero the output as and

frequencies were zero the output as and AI should also be zero meaning if I

AI should also be zero meaning if I don't put light in those two frequencies

don't put light in those two frequencies I should not see any light in those

I should not see any light in those frequencies and the pump stays the pump

frequencies and the pump stays the pump but in the lab then you do the

but in the lab then you do the experiment you actually see light in

experiment you actually see light in those signal Eiler frequencies

those signal Eiler frequencies okay something like what's happening the

okay something like what's happening the classical nonlinear Optics is not

classical nonlinear Optics is not describing it correctly

describing it correctly but then if you take those this coupled

but then if you take those this coupled mode equations connecting asai in an

mode equations connecting asai in an asai out

asai out you put hats on those A's and make them

you put hats on those A's and make them Annihilation operators then the whole

Annihilation operators then the whole thing becomes a two mode boogalube of

thing becomes a two mode boogalube of transformation or two more squeezing

transformation or two more squeezing transformation

transformation and we know that a two more squeezing

and we know that a two more squeezing operator acting on vacuum vacuum will

operator acting on vacuum vacuum will give you a two mode screen State yeah

give you a two mode screen State yeah so but then you might still ask

so but then you might still ask where did the photons come from you just

where did the photons come from you just told me that squeeze light has photons

told me that squeeze light has photons but I did not put any photons in those

but I did not put any photons in those that cannot happen physics thought allow

that cannot happen physics thought allow doesn't allow it so the photons

doesn't allow it so the photons obviously came from somewhere they came

obviously came from somewhere they came from the pump

from the pump but the pump is 10 to the six times

but the pump is 10 to the six times stronger than the than the squeeze light

stronger than the than the squeeze light that is coming so the pump depletion is

that is coming so the pump depletion is very negligible

very negligible or we assume it to be very negligible so

or we assume it to be very negligible so we say that we are operating in a

we say that we are operating in a non-depleted pump regime

non-depleted pump regime in that case you can think of that as

in that case you can think of that as squeezing the vacuum

squeezing the vacuum but if your pump is not strong

but if your pump is not strong then the depletion from the pump has to

then the depletion from the pump has to be properly taken into account yeah and

be properly taken into account yeah and then that three-mode hamiltonian gives

then that three-mode hamiltonian gives rise to very interesting non-gaussian

rise to very interesting non-gaussian characteristics that somebody bowling

characteristics that somebody bowling ski Carmichael and many others a Quantum

ski Carmichael and many others a Quantum Optics studied that you can generate

Optics studied that you can generate this interesting non-gaussian states but

this interesting non-gaussian states but the regime that most people in the

the regime that most people in the community use this is in the

community use this is in the non-depleted pump regime okay so the

non-depleted pump regime okay so the non-depleted pump regime will give you

non-depleted pump regime will give you the single Photon I mean usually spdc is

the single Photon I mean usually spdc is also used for generating single photons

also used for generating single photons well single that's a good so single

well single that's a good so single photons is a byproduct that people do

photons is a byproduct that people do afterwards all spdc sources in fact I

afterwards all spdc sources in fact I was visiting Professor varshi sinha's

was visiting Professor varshi sinha's lab this morning and we were discussing

lab this morning and we were discussing exactly this topic

exactly this topic every spdc and spontaneous photo mixing

every spdc and spontaneous photo mixing Source in the world what comes out is a

Source in the world what comes out is a two more space vacuum okay it says and

two more space vacuum okay it says and two more squeeze vacuum if you write

two more squeeze vacuum if you write down in the fork basis it's a

down in the fork basis it's a superposition of zero zero one one two

superposition of zero zero one one two two three three and so on

two three three and so on if you turn your pump power down your

if you turn your pump power down your two two three three four four are very

two two three three four four are very small so what people do is that they

small so what people do is that they stick a detector in one of the two of

stick a detector in one of the two of modes so if the detector clicks with

modes so if the detector clicks with high probability the other mode has one

high probability the other mode has one photon

photon so they use the spdc as a means to

so they use the spdc as a means to generate a single Photon other people

generate a single Photon other people use single Photon emitters like color

use single Photon emitters like color centers and Diamond to generate single

centers and Diamond to generate single photons to a totally different physics

photons to a totally different physics but spdc the raw state has always

but spdc the raw state has always squeezed light

squeezed light got it so thank you

got it so thank you right so I think we can take the rest of

right so I think we can take the rest of the questions out for coffee

interact with them we don't have too much time we are running a little late

much time we are running a little late uh Randolph Applause for

uh Randolph Applause for thank you we'll be back

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YouTube Transcript:Quantum Sensor Networks (Lecture 1) by Saikat Guha

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Quantum Sensor Networks (Lecture 1) by Saikat Guha