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Quantum Estimation for Quantum Technology

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the speaker today is M Paris he

associate professor at the department of

physics at the University of Milan and

he got his PhD in

94 uh the University of Pia he was then

for a couple of years the nonclassic

life group of the max Society in Berlin

and then back toia as a g dog and uh he

then moved to Milan and from one year

one year ago he was

a he

Associated um so he's working in Quantum

information and Quantum Optics he's

visiting us for almost one month he will

be here until the 9 of July so anyone

interest 10 days and thank you very much

and good afternoon thank you for being

here and you know already my name and

today I would like to give a brief

review of what is called local Quantum estimation

estimation

Theory and and then I will try to to

give some general results and then to

with some examples in compr examp

examp

and uh my title is quantum estimation

for Quantum technology because the

motivation for this line of research is

physically the fact that the quantum

mass of physical

systems is usually a resource to do

something better than without the

quantities and the examples are many

from from Quantum information to high

prision measurement and mography and

what they have in common all these

examples and then Quantum the quantum

feature are a resource and we need the

characterization the precise

characterization at the quantum level of

states and operations these are the

basic the general motivation of this

line of

research um what I'm going to speak is

something specific uh this is the

the general scheme I will deal with and

the idea is the following I have a

physical system which may be a single

Cub or a set of many illat and I assume

that this physical system may be

prepared in several different Quantum

States possible Quantum States and these

are here leled by by some parameter or

set of parameters L and the overall goal

of what I'm I'm going to speak about is

to find a way way to estimate the value

of this parameter which is the same to

discriminate which is the actual

preparation of myant system among the

possible different SE

prar and the this uh the proced is to

prepare several times and my system in

the same preparation and then to perform a

measure and Quantum estimation Theory

provides the the tool to individ way the

optimal measurement in a sense to be

specified in the following SL and in

turn to to say which is the which are

the ultimate bounds to Precision in the

estimation of this

paramet and the state the the fact that

my physical system is been prepared in

some state may be the result of the

preparation procedur so in this case I'm

dealing with the characterization of

quum states or may be the result of the

action of some device on a

fixed known initial St in this case I'm dealing

dealing

with operation the characterization of

operations but of course the two

problems may be M into each so this is

the the

basic the basic problem I would like to

estimate the value of the parameter

leling the state of a physical system

one may wonder that the answer to this

problem is already contained in the

postulat Quant mechanics after all we

have a physical quantity what we have

learned is to measure the corresponding

observ and the ultimate Precision in

measuring this observ is set by

relations and this is of course true but

then two remarks are in order and the

first is practical uh if you need to

measure energy from time to time your

physical system the energy is not

assessed so par reason because that kind

of measurement is not

implemented and the second remarks more

fundamental and there are several

quantities of

Interest which not even in principle May

correspond to an observ entropy fory

entanglement purity of quantum State all

these quantities are nonlinear functions

of the density of so not even in

principle they may correspond with

qu and other

back and other examples are time time is

not an

observ but it's not time observ there is

no phase of the quantum illat exception

no this examples even in principle I

cannot follow the standard procedure of

measuring the observable and then look

at the observative

relation so now again look at the same

problem perhaps it looks less Tri than

before so at least it is wor to to deal

with Quantum estimation with is adding

something to standard mechanics to

answer in the general case to this

problem of

estimating and to attack the problem

let's forget Quantum Mechanics for a

while and U and see what classical

statistics is telling us about this mod

so now I have no role is a physical

system no Quantum just a physical system

and for some reason I assume that I'm

interested in measuring some parameters

some quantity and I cannot assess by

direct measurement this one I have to

resort for some reason to an indir

measurement so I'm interested in Lambda

I'm measuring something

else the problem so I preper several

times my system in the same preparation

I perform uh the measurement I have an

experimental set different value of the

quantity X and from this experimental

sample I would like to estimate the

value of this problem the problem is

naturally splited in two parts the first

is the choice of the measurement you

estimate the quantity L and the second

is the way to

treat the the experiment so I have to

measure something I choose an observ and

then I can take the mean I can take the

sign of the mean I can take the cosine

of each value and then take the mean all

this function are called

sors so the problem is splitted in two

parts the first the choice of the

measurement the second the choice of the

SD there is a fundamental result in

classical statistics which is called the

cam B Cam theor and is saying that if I

have to estimate a quantity Lambda and I

the variance which express the Precision

of this estimat proced the variance of

any unbiased estimator is bounded by

this qu m is the number of measurement

which is just statistical S and F is the

So-Cal feature information which is buil

with the conditional distribution of

what I'm measuring X conditioning to the

true value of the T

so here we have the the answer to the

the problem on which is the best way to

estimate the

parameter um the optimal measurement is

the measurement which gives you the

maximum of the ficient information so if

you have to compare two different

measurement you have to choose the one

with the maximum fish information which

is quantifying the sensitivity of the

your distribution you have a given value

of the parameter you have a given

distribution if you move the value of

your parameter the icial information is

quantifying how far is set your

distribution and of course this is to

compare different measurements but then

this is an inequality

inequality

and there is a second part what is an

optimal estimator the optimal estimator

is the one

saturating the uh camera one so if you

have an estimation problem uh you have

first to choose the optimum measurement

maximum feure and there is no no way no

constructive way to to be in classical

physics to be the optimum measur and the

second is to find an optimal estimator

and again we have no theor to to find

the optimal estimator but at least we

have a couple of example

and and the first example is the biut

Once you have chosen a measurement and

then you have to choose the estimator

how to treat to process data and uh

hopefully with with the idea of

saturating the FR inequality and the

first estimator which at least in the

Astic region is always optimal which

means it saturates inquality is the bi

estimator if you measure the quantity

X conditioned to the value of the

parameter then you are interested in the

in the in the distribution of the

outcomes the experimental sample and you

follow the standard bian recipe and the

bian estimator is just the mean of the

posterior distribution there is a nice

theorem the plus theorem says that in

the asymptotic limit if you perform any

measurement the posterior distribution

is a gan a gaan with a variance which is

the saturation of the

camera so at least in the Astic region

there is always an an optimal

and there is another example which is

the maximum likelihood estimators and

again in the asymptotic region and you

have optimality in the sense of

saturation of the camer okay this is

what is known from classical

statistics you have to maximize the fish

information no no k no constructing way

to do this and then you have to find an

optimal estimator at least in thetic we

know that there

are now let's go one try to to reproduce

this result in the quantum and the

scheme is the same at the beginning but

now we have the B rule to to build the

conditional distribution we are

measuring an observable in general or

something general like a PM so people

know what the PM is and then we build

the probability density and

then we need to perform the derivative

in order to to build the deficient

information and this is usually done by

constructing the soal symmetric

logarithmic derivative which is a s

joint operate of the in this way with a

zero mean then you put everything in the

ficient information and you have thisch

in terms of the spectrum measure you are

performing on your system what is l l is

called a symmetric logarithmic

derivative is an operator defined

implicitly defined by this relation

of yes yes this what will be one

important one and the fact that it

depends on L give the name to local qu

estimation in I don't know if you know

something about this classical

estimation here we have two approach the

global looking for the best measurement

which is the optimal in average for all

the value of the parameters then you

have local Quantum estimation here and

then Quantum estimation here which is

looking for the op estimat for a given

value this appears to be a logical look

but I will speak about this okay and so

you will the the the ficial

bu by a set of inequalities which I will

skip immediately you arrive basically

the the CIS inequality in the in in the

space you having this nice very nice

inequality which say that the feature

information of any possible Quantum

measurement you may perform on a system

is bounded by this quantity which is

called the quantum fure

information so you have you know the

maximum of the effici the overall

maximum over the possible all all

possible measurements is given by this

quantity which depends only on the

family of quantum State you are

considering because broland is

explicitly there and the symmetric lar

derivative is built in

terms of the family of

ques so this is something which is

essentially Quantum because in classical

statistics there is no way

to find the maximum and there is not

even an indication of which is the

maximum in quantum mechanics there is a

way to find the maximum deficient

information and this is the maximum of

over all possible Quantum measurement you

you

make and so there is what is called

Quantum inequality the variance of any

Ed estimation is bounded by this one and

also there is a way to find the optimal

measurement achieving

this maximum and this is the

measurement for which the two

inequalities are saturated and I will

see the details if the measurement

satisfying this inequality in

terms the the unnown here the element of

the spectrum measure and you discover

that the optimal measurement is made of

the is the set of

projectors built with the alien state of

so over we have the answer you have the

family of quantum State you would like

to discriminate among these Quantum

States or which is the same to estimate

the parameter L the family of quantum

State and the answer is the optimal

measurement is the set of projectors

given by a state of the symmetric

logarithmic derivative and then this is

the measurement then you have to process

data but the processing of data is still

toxic the qu what we are quantizing is

just the measurement but then the the

post processing you may use uh whatever

uh estimator you you know is optimal in

this case at least we know that the B

estimator and the maximum life estimat

they are so this is the full solution uh

look for this then then we have model by

model uh we have to check if this is can

be implemented in practice but at least

we have the the complete

head and now as somebody was noticing

before the optimal measurement of the

quantity Lambda depends on Lambda itself

this appears to be a

logical and this is common is not it's

not essential Quantum is the point of

loc ofation Theory and how this can be

can be adapted the first is to perform

the feedback ass system measurement or

even simply more simply uh just

choose a measurement perhaps given by

the the measurement provided by global

Quantum estimation Theory have a rough

estimate of the parameter on a small

fraction of copy you have a disposal and

then perform the optimal measurement

because you have an idea or which is the

value of the parameter and then you may

implement the measurement of

given by this recipe of course you have

to be sure that this

iterative procedure is converging and

for this there are no differ and you

have to check model by mod and

fortunately enough all the examples that

we have investigated this convergence is

ur but for this we are not human and

then there was another a possible question

question

about um all this estimation procedure

is performed with the assumption that we

have several repeated preparations of

our Quantum State we perform sequential

measurements one may wonder whether

entangled measure on on a global

preparation may be more convenient at

least for cubits this is not true for

this we have a theor proved by g m in 2000

2000

that there is no gain in using a

time and so for the estimation of a

single parameter uh this is the global

Optimum even

considering uh enti

okay so how to solve an estimation

problem you have the family of quantum

State at first you have to solve this

equation to find the symmetric

logarithmic derivative and then plug

into the quantum FAL information this is

is telling you which is the best

measurement and this is telling you

which is the ultimate Precision to the

estimation of the these are operatorial

solution because this is a y like

equation but of course in practice you

almost never using this this

construction what you you use it force

it to do

is to diagonalize the family of quantum

State both the the value and the vectors

depends on the quantity Lambda and then

you plug this in the previous formula

you have General formulas for the

symmetric lari and the quantum fish

information this is this is nice because

you have two pieces

and the the idea is to maximize the

fishal information and this is called

the classical part of the quantum ficial

information this is called the quantum

part why because this is is measuring

how the a values of your of your state

depends on the quantity of L and this

is capturing the dependence on the

vectors now some example perhaps clarify

some some of the meaning of this so the

simplest example unitary family of

quantum State you have some known State

then you are pering the state with a

unitary peration this is the generator

of the perturbation this is the quantity

of interest if G is antonian this is

time if G is the number of photons this

is a phase and so on and so a shift a

general shift

parameter and the symmetric logarithm

derivative is covariant the problem is

covariant by definition and so this is

the General expression and this is the

general expression from the quantum F

information there is no classical part

because the peration is unitary and the

quantum Fusion information is

independent on the value of the

parameter because of the coal so you can

measure with the same Precision all the

possible value of the now let's write

the car bound uh and the first example

when the the initial state is view if

you perform the calculation when the

initial state is pure you end up with

the fact that the quantum ficial

information is four times the

fluctuations of the generator you plug

this into the camera around and you

arrive at this relation this is quite

appealing for a physicist because as a

firm of an uncertainty relation but this

is not an uncertainty relation this is

the cam this is an operator but this is

a general

parameter so um perhaps you have seen in

the literat some aristic uncertainty

relation for time or for phase and uh

they cannot be prop uncertainty relation

because they they are not observable and

what is uh doing for us the camer bound

is to go to give a firm statistical

ground to

thisis because this is very also when

Lambda is not does not correspond to an

Ober and when the initial state is mixed

we have two pieces this is the same as

before the this piece and then we have

something which depends on the F on the

fact that we are dealing with M also

this is nice this is not been pursued in

the literature but this is nice anyway

and because it is an answer to the long

standing debate about the origin of

noise in a measurement uh which is the

part of the noise depending on Quantum

fluctuation and which is the part of the

noise coming from classical fluctuation

here we have natural way to separate the

two contributions this is when the

initial sa is pure so this

is quantum Mon and this is the

statistical fluctuation in the

preparation okay then when you have a

your Quantum fure information you may be

the signal to no ration to to quantify

in way the estimability of the parameter

and this is the ratio between the value

of the parameter and the noise and of

course the signal to noise of any

possible measurement is bounded by what

we may call the quantum signal noise ra

and this you may be the several quantity

as the number of measurement to achieve

some some some even relative

combinations different figures of Merit

to to quantify

the estimability of the parameter and by

this you discover that for example if

you have a a unitary parameter and then

we have already discovered that the

quantum F information is independent on

the value of parameter and so the the

the quantum signal is is Vanishing for

Vanishing which means that three thing

if you have a parameter and the

Precision is constant

uh well not depending on whether the the

parameter is large or small then this

means that this parameter is

in badly estimated because when it is

small the error it remains large but

this is not a general is a feature of VI

T but it is not a general feature I have

an example and this is my first example of

applications and supposed to

have and illat which is modeling the

light a single model the ration field

and you put your uh your light beam in a

noisy Channel which means the the

dynamical equation is the master

equation you put your oscillator in

interaction with the bath of them

oscillator at zero temperature and theal

evolution is this master equation where

this is the relation with The Dumping of

the ch and this is the blood operator

and if you are interested in

characterizing the channel so in

estimating the loss parameter what you

can do is to prove to send some some

Pro State and then to perform

measurement on the output State this is

nice because this is a good model for

absorption of light beam and for

propagation with noise each at least

when the temperature is very low you

don't have teral contribution if you

apply the Machinery of quantum

estimation Theory you discover that the

variance of the optimal estimator which

is achieved when here you perform some

some photon number detection and some

gaan operation is bounded by this one

and this is nice because the cam bound

says that the variance is proportional

to the loss parameter itself so also

when the loss parameter is very small

you may hope

to estimate the L with high

position because the the the

noise is is going to zero together with

the value of the time this is not

countering think of the probability of

flipping a coin the variance of of

thean distribution and the variance of

the event is going to zero together with

probability is a similar to

this then another example and the

estimation of the phase in the presence

of pH diffusion here the

parameter is unitary again a single mode

radiation field and you interested in in

estimating the pH sheet perhaps coming

from from a different Optical path in

Teeter and you assume that uh there is

some noise phase diffusion gausian phase

diffusion here so that the measurement

scheme is the following you have an

initial State a know initial State then

you perform the phase shift your inter

meter here and then perhap some noise

before the measurement but you may

exchange this because the noise and and

the and the perturbation

they and when there is no noise the

maximum of the of the ficial information

is known to be achieved using here soal

squeeze State The Squeeze vacuum and is

proportional to the square of the energy

so the

the uncertainty in the in the estimation

of the phase may be May decrease one

over and square when n is the number the

mean number of What's happen when the

you have a phase phase diffusion and of

course from the form of the noisy

channel of the phase diffusion you

expect some dependence on on the square

root of En on the square of energy

multiplied by the noise parameter and

this is captured by by the analysis with

with efficient information because we

discover a scaling load for the fish

information and for the squeezing

fraction U when you have no

noise The Squeeze infrction is one

because the op squeeze vacuum and then

when you have noise you cannot

appreciate the advantages of squeezing

it so the squeeing fraction is

decreasing and and there other results you

you

discover with with a tool from Quantum

estimation Theory and to perform a Bine

detection measure the field quadrature

at the output is nearly optimal at least

when the noise is very small was very

high so in this case uh the optimal

measurement cannot be implemented but at

least you have a benchmark to judge what

can be done in qu which is detection

basically or or photo detection which in

this case is not is not use

now uh for the I mentioned at the

beginning that this may be applied also

when the set of

parameters when the states are led by a

set of parameters rather than a single

parameter so there is an extension to

the multiparameter case it's quite

straightforward instead of of the fure

information you have the quantum fish

information metrix and the elements are

buil uh with

building and a symmetric logarithmic

derivative for each

parameter and the analog of the bound is

given for the ciance of the estimator in

general this inequality is not

achievable is not ATT is a real is a

matrix inquality so cannot be achieved

in general but if you have a

multiparameter problem and you are

interested in the estimating a single

parameter and fixed value of the other

parameters then you have this

generalized PR and this is achievable in

and why I'm mentioning this because this

is the typical problem when you want to

estimate entanglement

of in general you you would like to

estimate entanglement but there are

other parameters governing States Other

M Elements which are relevant for the

preparation of your Quantum state so you

have just this this kind of problem and

to introduce the problem of estimation

entanglement just mentioned the simplest

case you have a pure State and we have

here we have a single parameter and the

entanglement is a monot with q and in

this case um you apply the Machinery of

quantum machine Theory and you arrive at

this expression for the the quantum

signal to no solution and the meaning of

this is uh is here

and the quantum fure information is

increasing with enang lement and

diverges for maximal which means that if

you have a family of sa then when theang

is very large close to one to Infinity

depending on the range of your measure

and then can be estimated with high

Precision this is a good news because if

you are interested in in a device

preparing a tangle thing then of course

you would like be a to be large because

the resource and so you have your device

preparing the state you would like to

check if you have prepared maximum St or

not your is large and you perform the optimal

optimal

measurement VI by by by Quantum

estimation Theory and you expect an

estimation very

precise and so you may improve your

generation scheme because the estimation

is very

conversely when the anang is low then

the quantum information is low and so

estimation of is inherently inent so the

error in estimating the value is very

large when theang is and this is quite

General feature depends on the dimension

of space whether you have Cubit cued or

continuous variables but the general

feature is that when the is larger can be

estimated very you know very that is

valid for any property this

green not only for

for

um estimation of a small values of anything

anything

is well not for for the unitary case

definitely but when I was speaking about

the the L parameter of the channel then

was the opposite the noise was scaling

the parameters so in that case that's not

not

true uh usually you have fixed the

variance of your exat SK set so of

course if the is more this is but this

is the was not granted from the very

beginning because this is linear

function of the so I would say this is a

result I see

unfortunately I would have preferred

another result when the can be estimated

in a good way the whole range but this

is not the case

and this is the experimental

implementation of this made at National

metrological Institute in in Torino they

are able to prepare this family of State

this is a maximally entang state of two

can be a maxim state of any any uh

superposition of two photons of vertical

andiz polarization so this is a the

polarization in States so they they can

tune uh both the

parameters the the balance of the two

components here and the mixing parameter

because what do you have in a

real device for preparing Timeless state

is something mixing the the ESS state

with a mixture

mixture

and what we had done is to look for an

estimator a real estimator built with

some some interference and visibility

measurement and what we have discovered

that optimal estimation is

possible and if you perform some

combination here of visibility

measurement you have an estimator and

this estimator is saturating the cam so

optimal estimation of anang is

possible and the result are here so here

is the estimated value versus the true

value of

entanglement obtained because the Peter

can be estimated independently and um

and in fact we have an error also on the

true value and the gray area is the

camera B and the blue points and line

are the the the experimental

determination together with the ARs so

the results is saying that optimal

estimation of entanglement is possible

but of course when the entanglement is

very large the estimation is very good

which means that the AR bars are small

and then when en is small the the the

estimation here bu very so but this is a

a property of entanglement so the good

news is that optimal estimation is

possible and the bad news that

entanglement is

inherent uh inefficiently

inefficiently

estimated then my few last

slides a connection with the quantum phase

phase Transitions

Transitions

and so in General the phase transitions

in a physical system is associated with

some strong change in some Rel Observer

so you connect the change the state of

the system with the change of some microscopic

microscopic

parameter and Quantum in Quantum phase

transition uh you expect that to have a strong

strong

departure from the initial Point density

mat so if you have the ground state of

the system and you say and you say that

this system is experiencing a Quantum

transition you expect the ground state

to change

very and you expect this to be

Quantified in term of some distance

between states in the

interace and uh this approach was

followed by by zalan who works at the

isi foundation few years ago to rephrase

to consider the problem of identifying

one to transition instead of

using a microscopic parameter use a

distance between

Quantum some kind of geometric approach to

Quantum and for example you may quantify

this by soal burus distance between

Quantum SES which is built in terms of

fidelity if you consider two uh ground

stes of a g even many body system and

you change the parameters of the

interaction of the preparation or the

temperature whatever then you may

quantify the distance between the two

different ground States infinitesimally

close ground states by by the tensor

metc of of the L

distance now why this is connected qu

estimation because this the answer is in this

this

and when you have a set of parameters

classical set of parameter you may IND

use a statistical

differential manifold by considering the

measurement to estimate the set of

parameter and the fure metric in the

multiparametric case I was mentioning is

providing a metric the fish Quantum

fure Matrix is providing a metric for

this statistical mag but the same can be

done if you map the set of parameter

into a family of qu St this was

our initial starting point to Define quantum

quantum

estimation and here in the manifold of

quantum State you have distances and the

connection in between the two approaches

is just the optimization if you optimize

of all possible Quantum measurement you

are performing here then you arrive at

the quantum fish metric and the result

is that the quantum fion metc is just

proportional to the Tens metc coming

from the distance

so yes just as calculation what is

saying is that the buis metric is the

among the different Matrix you may

Define on the manifold Quantum State the

BL Matrix is capturing the statistical

distinguishability the other distan is

perhaps capturing something else but the

statistical distinguish term of

measurement is captured by the

the

and this is telling you something which

is the role quite trivial that if you

have a parameter leveling the ground

state of a b body system then the

estimation of this parameter is very

effective at critical points this is

something it sounds but then we have

some some analytical proof of of this

intuition and in fact you May I'm

skipping details but you may perform

analysis on some super extensive

behavior of the of the of the quantum

ficient information U respect to the

size of the system of the temperature uh

say I mean quantify this this this

statement that estimation is very

affecting a critical point just an

example before before going to the end

take the the simplest example the easing

model with the transer transverse

field and suppose you would like to to

estimate the cing Conant you have a

chain of spins and you want to

characterize the system which means to

to estimate the value of the

C and what you discover that the quantum

Fusion information at zero temperature

is maximized at the P point for any this

is the result because for final size

this is not a critical point what is

called the critical points you have only

the teram but here the result is that

even when the size is fin the quantum

fishal information is maximized at a

same the critical point and you have a

super extensive behavior of the quantum

feure information and another nice

results I would say is that one may

Wonder but the the measurement achieving

this Quant ficial information is visible

in general no is a licated expression in

terms of the symmetric L derivative does

not correspond to something may be

implemented in practice what you can

measure on a spin chain is the the total

magnetization this can be measured and

what what you discover as from the the

ratio the fishal information of this

measurement to the quantum fishal

information then at least when the

coupling is not too small the ra is one

so also in this case in some regimes you

have that the C BS can be achieved so

optimal estimation can be can be

achieved so sumary

sumary

um let's make this

short um Quantum estimation is a tool

for Quantum technology because there are

several quantity of Interest which are

not corresponding to obser so we need

something more that hos quantum

mechanics to attack the problem

and the recipe is quite clear and you

have also benefit of being Quantum that

you know the maximum efficient

information and you know you have the

full recipe to arrive at the optimal

estimation and either to implement the

optimation scheme or to use it as a

benchmark to to judge what can be

implemented on your

system and and then there are few few

few more results about the theability of

the parameter

and about this is me not explored I have

no time to present result about the the

the classical and Quantum contribution

much a good measure of

this noise

ratio is the Mand parameter Factor

can you relate that to to the quantum

information some connection

connection

um at least yes well of course on a

restricted set of physical

system basically the radiation Fe yes

where the final Factor has an immediate

meaning in terms of classicality and

classicality yes in that case yes in

general uh in general no because the

final factor for a fin dimensional

system is not immediately related to any

Quantum so but for the radiation field

for the radiation field um yeah

definitely the answer is yes I don't

know the complete answer but I'm quite sure

they question

question

I like just at the very beginning just to

clarify you CH a physical meaning just

repeating you already told about this

inequality you pass okay about this F of

is okay the fish the fish this is a this

is the find

here yes with theal probab yes so the

ficial information can be written

evaluated once you have chosen a

measure I all the possible measurements

you may perform on a system you choose

one and then the uh the

Precision in the estimation of a given

parameter starting from that

sample measurement is ruled by the

camer you want to maximize

this and

the other inequality the quantum

inquality says

that choose one measurement and choose

another one you compare that but the

ficial information of any measurement

you may perform a Quantum system is

anyway bounded by this

qu so if you choose a measurement whose

FAL information is equal to the quantum

F information then you you cannot do

so it's just saying

that um this is nice

because basically the set of quantum

measurements is well

characterized you have S Spectrum

measure of

Po uh the set of classical measurements

performed on a physical system is not well

well

characterized how to what what is a a a

measurement in classical physics is a

distribution but then a distribution in which

which

space uh which constraints you put on on

your distribution to be physically

admissible it's not completely clear

there is no a classical theory of

measurement in this sense in quantum

mechanics we have quantum theory of

measurement and we have the full characterization

characterization

p and this fact um made possible to

write down two to obtain the the the the

ultimate maximum of deficient

consequence can I

have can you say something more about

yes of course

here of course uh this is Val for

is so this is the starting point and

then you ask about the contribution to

noise so when the state is pure

everything is quite clear and this is

what you obtain when the state is

it's nice because we have a

sum but it's bad

because this superator form is not easy

to enter this is why everybody is saying

very nice but then nobody's working with

this subject because nobody found a way

to deal with all this

operatorial and so on and so for so in

principle the answer is there but then

thank

you just the last you have shown this

you this you expect anything s for continu

variables just a very specific results

no yes and no we already um did on

paper the attack the problem we know we

know the bounds we know

everything but then in that case

the what can be done in a lab for the

moment is not achieving the

maximum so you have a way to estimate

entanglement is much more noisy than the

ultimate B

provided by by the

C so for the moment of course enang for

can be estimated but nobody found a way

Sate because the set of possible

measurements on the continuous variables

say are is restricted I mean you have an

infinite number of parameters and then

you may just count photons or measure of

the field no

they any other

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