BONA FIDE MEASURES OF NON-CLASSICAL CORRELATIONS · BONA FIDE MEASURES OF NON-CLASSICAL...

BONA FIDE MEASURES OF NON-CLASSICAL CORRELATIONS

New J. Phys. 16, 073010 (2014)

A. Farace, L. Rigovacca and V. Giovannetti

A. De Pasqualein collaboration with

http://arxiv.org/abs/1305.3069

http://arxiv.org/abs/1305.3069

Outline

MAIN IDEA: Introduction of measures of non-classical correlations

classical and non-classical correlations

discord, as difference between two definitions of mutual information

Discriminating Strength (DS)

brief review of some measures of discord, in the context of quantum metrology

Correlations: classical and quantum

A

B

Classical Correlations

Quantum Correlations

Two systems are correlated if together they contain more information than taken separately

Can we establish the nature of such correlations?

We measure correlations in terms of experimental observations on both subsystems

inspired by G. Adesso’s seminar at "46 Symposium on Mathematical Physics: Information Theory & Quantum Physics", Torun 2014

Classical correlations

I(A : B) = H(A) +H(B)−H(A,B)

J(B|A) = H(B)−H(B|A)

we measure A

=

H(A,B)

H(A) H(B)

I(A :B)

H(B|A

)

Bayes’ rule pb|a =pabpa

H(B|A) =�

a

paH(B|a)

MUTUALINFO

Nielsen & Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press

pa =�

b

pabA,B random variables with joint probability distribution , ,pab pb =�

a

pab

Shannon entropyH(A) = −�

a

pa log pathe amount of information we gain on average when we learn the value of A

Correlations and Entanglementent

A

B

INTERFERENCE among probability amplitudes

pure states: entanglement quantum (non-classical) correlations⇐⇒

|ψ� =�

i,j

cij |ψi�A ⊗ |ψj�B �=|ψ�A ⊗ |ψ�B

“the best possible knowledge of a whole does not include the best possibleknowledge of its parts, even though they may be entirely separated”

Entanglement, Schrodinger 1935

:

A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935); E. Schrodinger, Natu !rwissenschaften 23, 807 (1935)

Correlations and Entanglementent

A

ensembleof pure states {pi, |ψi�}

mixed states: entanglement quantum correlations⇐⇒?

ρ =�

i

pi|ψi��ψi| �=�

k

p̃k ρkA ⊗ ρkB

B

“the best possible knowledge of a whole does not include the best possibleknowledge of its parts, even though they may be entirely separated”

Entanglement, Schrodinger 1935

:

incomplete knowledge of the system(e.g. uncontrolled interactions with the environment)

R. F. Werner, Phys. Rev. A 40 (1989) 4277; R. Horodecki, M. Horodecki, Phys. Rev. A 54, 1838 (1996)

=

we measureA

J(B|A) = maxΠA

j

�S(ρB)− S(B|{ΠA

j })�

I(A : B) = S(ρA) + S(ρB)− S(ρ)

MUTUALINFO

von Neumann entropy

S(ρ) = −Tr[ρ log ρ]H(A) = −�

a

pa log pa

Shannon entropy

AB a bipartite system described by the density matrix ρ →ρA = TrB [ρ]

ρB = TrA[ρ]

Quantum correlations

Discord: purely quantum correlations

For pure states (i.e. ) discord is a measure of entanglement

There are separable mixed states (i.e. ) such that D(B|A) > 0

ρ = |ψ��ψ|

ρ =�

i

piρAi ⊗ ρBi

CQ statesD(B|A) = 0 ⇐⇒ ρ =

�

j

pj |j�A�j|⊗ ρBj

D(B|A) = I(A : B)− J(B|A) ≥ 0

all possible correlations

between A and B

orthonormal basis{|j�A}

measures on A: classical fraction of correlations

ρentρent

ρdiscρdiscρclρcl

L. Henderson and V. Vedral, J. Phys. A: Math. Gen. 34 6899 (2001); H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88 017901 (2002)

ρsepρsep

How to measure discord?

D(B|A) = S(ρA)− S(ρAB) + minΠA

j


j })� difficult

to compute

geometric discordtrace distance discorddiscord of responsemeasurement-induced disturbancemeasurement-induced non-localitylocal quantum uncertaintyinterferometric power...

QUANTUM ILLUMINATIONAND METROLOGY

?

?

??

S. Lloyd, Science 321 (2008)

A

2

A

B

C

D

E

!"

C

A

B

D

!"E

FIG. 1: Blind quantum estimation. TOP: Alice and Bob initialize the

two arms A and B of an interferometer in a probe state ρAB . Alice’s

subsystem undergoes a unitary dynamics described by UA = e−iϕH

A,

where ϕ is the parameter to be estimated, while the Hamiltonian HA

is secretely determined by Charlie (C) who reveals his choice only

after the probe state has been transformed. Alice and Bob are then

asked to retrieve ϕ upon performing the most informative joint detec-

tion (D) on the output state and constructing the best estimator ϕ̃ (E).

If ρAB is uncorrelated or only classically correlated, it is impossible

to guarantee a successful estimation for all possible moves of Char-

lie. Exploiting instead probe states with nonclassical correlations

(with or without entanglement), Alice and Bob can always estimate

ϕ with nonvanishing precision. The worst-case precision defines the

interferometric power P A

of ρAB , which is a measure of its quantum

discord. BOTTOM: Remote sensing application. A satellite encodes

a message in a phase ϕ. Upon receiving a probe signal, the satellite

bounces it back shifted by ϕ in a direction �n. For security reasons,

the direction is randomly changed after each time interval δt, and

then publicly broadcast. If δt is smaller than the time needed for a

signal fromearth to reach the satellite, then the actual �n which will

be applied is totally unknown at the state preparation stage, realiz-

ing an instance of blind metrology. This is enough to prevent purely

classical players from gaining any information about ϕ in the worst-

case. Conversely, any state preparation making use of discord always

ensures a nonzero minimum precision, quantified by P A(ρ

AB).

determine as precisely as possible an unknown phase ϕ intro-

duced by an assigned black box device whose unitary phase-

imprinting mechanism, generated by HA , is unknown at the

state preparation stage of the input probe. Think for instance

to a satellite interrogation (Fig. 1) or a quantum illumination

setting [10] where Alice is asked to monitor a remote (unco-

operative) target whose interaction with the probing signals is

partially incognito. Let us first consider the case of unassisted

probing (i.e. no reference system B). Alice equips herself with

a qubit probe initialized in a state ρA of her choice. The probe

enters the black box, where a randomizing mechanism, or an

intelligent referee called Charlie, decides the direction �n on

the spot and rotates the probe by ϕ according to the generator

HA = �n · �σ

A . Charlie can now disclose the chosen setting �n to

Alice, who recovers her rotated probe and implements the best

possible measurement strategy to estimate ϕ. The trial can

be repeated an arbitrarily high number ν of times to improve

the statistics, under the condition that the prepared quantum

state ρA and the Bloch sphere direction �n are fixed by the first

trial and not changed during the whole procedure. Eventu-

ally, Alice deduces a probability distribution for ϕ; the esti-

mation precision shall be determined by the associated QFI.

How can Alice choose a probe state ρA that guarantees her a

nonzero precision whichever the setting? Simply, she cannot,

as for any ρA there are always adverse choices of �n such that

her state is unaffected by the rotation, resulting in a zero QFI,

or not sufficiently affected for the task purposes, resulting in

Alice being unable to access information about ϕ precisely

enough. The minimum precision over all �n vanishes as it is in

fact impossible for a qubit state ρA to exhibit coherence in the

eigenbases of all Hamiltonians �n · �σA .

The solution to this conundrumrequires a collaborative

strategy based on the interferometric setup of Fig. 1. Alice

and Bob initialize qubits A and B in a chosen probe state ρAB ,

unbeknownst of �n. As usual, after Charlie discloses �n at the

output stage, Alice and Bob are allowed to performthe best

possible joint measurement on the resulting global state ρ ϕAB ,

possibly repeating the estimation trial ν times. It is natural to

assign a relevant figure of merit for this procedure given by

the worst-case QFI over all possible black box settings �n,

P A(ρ

AB) = 1

4minH

A

F(ρAB ; H

A) ,

(1)

where we inserted a normalization factor 1

4 for convenience.

We shall refer to P A(ρ

AB) as the interferometric power (IP)

of the input state ρAB , since it quantifies rather intuitively the

guaranteed usefulness of such a state for blind estimation of a

phase applied on Alice’s side of the quantum interferometer.

All the states ρAB with nonzero IP are, by definition, useful

for blind phase estimation. Having already established that

product states are not in this class, one might wonder whether

entanglement between A and B is required for the task. Cru-

cially, we find that even the majority of mixed separable states

have a nonzero IP. Entanglement is not necessary to ensure

local coherence in all bases, but quantum discord is [11–13].

Discord encodes a statistical relationship between constituents

of a composite systemwhich has no classical analogue and

can be observed in the disturbance induced on the system

state by local measurements [7, 8]. While it has been spec-

ulated that discord might be at root of some quantum advan-

tage e.g. in specific computation or communication settings

[14–17], its practical merit remains unclear. We show that the

IP of Eq. (1)—which can furthermore be computed in closed

form for relevant cases [9]—is in general an operationally mo-

tivated and mathematically sound measure of discord. Dis-

?

MEASURE: well-defined, easy to compute, with a clear operative meaning

B

equality reached on pure states, where no classical igno-rance occurs (see Fig. 2). Hence, we adopt the skew infor-mation as a measure of quantum uncertainty and deliver atheoretical analysis in which we convey and discuss itsoperational interpretation.

As a central concept in our analysis, we introduce thelocal quantum uncertainty (LQU) as the minimum skewinformation achievable on a single local measurement. Weremark that by ‘‘measurement’’ in the following we alwaysrefer to a complete von Neumann measurement. Let ! !!AB be the state of a bipartite system, and let K!"K!

A # IBdenote a local observable, withK!

A a Hermitian operator onA with spectrum !. We require ! to be nondegenerate,which corresponds to maximally informative observableson A. The LQU with respect to subsystem A, optimizedover all local observables on A with nondegenerate spec-trum !, is then

U!A $!% ! min

K!I$!; K!%: (2)

Equation (2) defines a family of !-dependent quantities,one for each equivalence class of !-spectral local observ-ables over which the minimum skew information is calcu-lated. In practice, to evaluate the minimum in Eq. (2), it canbe convenient to parametrize the observables on A as

K!A " VAdiag$!%Vy

A , where VA is varied over the specialunitary group on A. In this representation, the (fixed)spectrum ! may be interpreted as a standard ‘‘ruler,’’fixing the units as well as the scale of the measurement(that is, the separation between adjacent ‘‘ticks’’), while VA

defines the measurement basis that can be varied arbitrarilyon the Hilbert space of A.In the following, we prove some general qualitative

properties of the !-dependent LQUs, which reveal theirintrinsic connection with nonclassical correlations.A class of quantum correlations measures.—What char-

acterizes a discordant state is, as anticipated, the nonexis-tence of quantum-certain local observables. In fact, we findthat each quantity U!

A $!% defined in Eq. (2) is not only anindicator but also a full-fledged measure of bipartite quan-tum correlations (see Fig. 1) [21]; i.e., it meets all theknown bona fide criteria for a discordlike quantifier [10].Specifically, in the Supplemental Material [22], we provethat the !-dependent LQU (for any nondegenerate !) isinvariant under local unitary operations, is nonincreasingunder local operations on B, vanishes if and only if ! isa zero discord state with respect to measurements on A,and reduces to an entanglement monotone when ! is a purestate.If we now specialize to the case of bipartite 2& d

systems, we further find that quantifying discord via theLQU is very advantageous in practice, compared to all

A B

(a)

(b)

A B

FIG. 1 (color online). Quantum correlations trigger local quan-tum uncertainty. Let us consider a bipartite state !. An observeron subsystem A is equipped with a quantum meter, a measure-ment device whose error bar shows the quantum uncertaintyonly. (Note that, in order to access such a quantity, the measure-ment of other observables that are defined on the full bipartitesystem may be required, in a procedure similar to state tomog-raphy) (a) If ! is uncorrelated or contains only classical corre-lations [darker (brown) inner shade], i.e., ! is of the form! " P

ipijiihijA # "iB (with fjiig an orthonormal basis for A)[8–10], the observer can measure at least one observable on Awithout any intrinsic quantum uncertainty. (b) If ! contains anonzero amount of quantum correlations [lighter (yellow) outershade], as quantified by entanglement for pure states [5] andquantum discord in general [10], any local measurement on A isaffected by quantum uncertainty. The minimum quantum uncer-tainty associated to a single measurement on subsystem A can beused to quantify discord in the state !, as perceived by theobserver on A. In this Letter, we adopt the Wigner-Yanase skewinformation [16] to measure the quantum uncertainty on localobservables.

0 14

12

34

10

12

1

p

loca

l unc

erta

inty

FIG. 2 (color online). The plot shows different contributions tothe error bar of spin measurements on subsystem A in a Wernerstate [5] ! " pj#'ih#'j' $1( p%I=4; p 2 )0; 1*, of twoqubits A and B. The solid red line is the variance Var!$"A

z % ofthe "A

z operator, which amounts to the total statistical uncer-tainty. The dashed blue curve represents the local quantumuncertainty UA$!%, which in this case is I$!;"A

z % (any localspin direction achieves the minimum for this class of states). Thedotted green curve depicts the (normalized) linear entropySL$!% " $4=3%$1( Trf!2g% of the global state !, which mea-sures its mixedness. Notice that the Werner state is separable forp + 1=3 but it always contains discord for p > 0.

PRL 110, 240402 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending14 JUNE 2013

240402-2

BA

minimum quantum uncertainty associated to local observables

zero q. uncertainty iff zero discord

UΛA→B

(ρ) = min{HΛ

A}I(ρ, HΛ

A)

LocalQuantum

Uncertainty

Skew Info. −1

2Tr{[√ρ, HΛ

A]}

D. Girolami, T. Tufarelli and G. Adesso, Phys. Rev. Lett. 110, 240402 (2013)

DR(ρ) = minUA

1

2D2

Bu(ρ, UAρU†A)

Discordof Response

Bures distance

local unitary whose eigenvalues are the roots of the unity

faithful measure of non-classical correlation, which generalizes the ENTANGLEMENT OF RESPONSE for pure states ER(|ψ�) = 1−max

UA

F (|ψ�, UA|ψ�)ER(|ψ�) = 1−maxUA

F (|ψ�, UA|ψ�)

2

A

B

C

D E!"

CA

B

D

!"E


two arms A and B of an interferometer in a probe state ρAB. Alice’s

subsystem undergoes a unitary dynamics described by UA = e−iϕHA ,











interferometric power PAof ρAB, which is a measure of its quantum






signal from earth to reach the satellite, then the actual �n which will





ensures a nonzero minimum precision, quantified by PA(ρAB).



imprinting mechanism, generated by HA, is unknown at the











HA = �n · �σA. Charlie can now disclose the chosen setting �n to

















eigenbases of all Hamiltonians �n · �σA.

The solution to this conundrum requires a collaborative


and Bob initialize qubits A and B in a chosen probe state ρAB,


output stage, Alice and Bob are allowed to perform the best

possible joint measurement on the resulting global state ρϕAB

,




PA(ρAB) =

1

4min

HA

F(ρAB; HA) , (1)

where we inserted a normalization factor1

4for convenience.

We shall refer to PA(ρAB) as the interferometric power (IP)

of the input state ρAB, since it quantifies rather intuitively the











of a composite system which has no classical analogue and









B

AUA

W. Roga, S.M. Gianpaolo and F. Illuminati, J. Phys. A: Math. Theor. 47, 365301 (2014)

Measures of discord related to q. metrology

D. Girolami et al, Phys. Rev. Lett. 112, 210401 (2014)GOAL:estimation of a continuous parameter ϕ

InterferometricPower

Quantum Fisher Information

PA→B(ρ) =1

4min{HΛ

A}F(ρ, HΛ

A)

2

A

B

C

D E!"

CA

B

D

!"E


two arms A and B of an interferometer in a probe state ρAB. Alice’s

subsystem undergoes a unitary dynamics described by UA = e−iϕHA ,











interferometric power PAof ρAB, which is a measure of its quantum






signal from earth to reach the satellite, then the actual �n which will





ensures a nonzero minimum precision, quantified by PA(ρAB).



imprinting mechanism, generated by HA, is unknown at the











HA = �n · �σA. Charlie can now disclose the chosen setting �n to

















eigenbases of all Hamiltonians �n · �σA.

The solution to this conundrum requires a collaborative


and Bob initialize qubits A and B in a chosen probe state ρAB,


output stage, Alice and Bob are allowed to perform the best

possible joint measurement on the resulting global state ρϕAB

,




PA(ρAB) =

1

4min

HA

F(ρAB; HA) , (1)

where we inserted a normalization factor1

4for convenience.

We shall refer to PA(ρAB) as the interferometric power (IP)

of the input state ρAB, since it quantifies rather intuitively the











of a composite system which has no classical analogue and









e−iϕHA

?






U!A $!% ! min

K!I$!; K!%: (2)


K!A " VAdiag$!%Vy







A B

(a)

(b)

A B



0 14

12

34

10

12

1

p

loca

l unc

erta

inty


z % ofthe "A




240402-2

BA

minimum quantum uncertainty associated to local observables

zero q. uncertainty iff zero discord

UΛA→B

(ρ) = min{HΛ

A}I(ρ, HΛ

A)

LocalQuantum

Uncertainty

Skew Info. −1

2Tr{[√ρ, HΛ

A]}

D. Girolami, T. Tufarelli and G. Adesso, Phys. Rev. Lett. 110, 240402 (2013)

D(B|A) = S(ρA)− S(ρAB) + minΠA

j


j })� difficult

to compute

local measurements on A

min/maxover the special

unitary group on A

Skew Info.

LQUUΛA→B

(ρ) = minH

ΛA

I(ρ, HΛA)

Quantum Fisher Info.

IP PA→B(ρ) =1

4minH

ΛA

F(ρ, HΛA)

Quantum METROLOGY

DR

Bures distance


(1) Alice: n copies of + Robert: a set of unitary rotations

(2) Alice sends the probing subsystems A to Robert while keeping the reference subsystems B

(3) Robert: apply or not apply ?

(4) Subsystems A back to Alice + Robert reveals

(5) Alice via optimal POVM on the n copies of AB: or

The discriminating strength

ρ⊗n (RAρR†A)

⊗n ?RA

RA

S={RA, R�A, R

��A, ..}ρ

RA

IAIARA. . . ˜ρ

}ρ⊗n

˜ρ ˜ρ(1)

(2)

R��A

R�A

(IAρIA)⊗n (p = 1/2)

(R?AρR

?†A )⊗n (p = 1/2)

(3)

(4)S

. . .

. . .

RobertAlice

A

B


RA

IAIARA. . . ˜ρ

}ρ⊗n

˜ρ ˜ρ(1)

(2)

R��A

R�A

(IAρIA)⊗n (p = 1/2)

(R?AρR

?†A )⊗n (p = 1/2)

(3)

(4)S

A

B

. . .

. . .

RobertAlice

QUANTUMCHERNOV BOUND

=

Alice aims to discriminate,via optimal POVM, between

andρ⊗n(RAρR

†A)

⊗n

GOAL:

P (n)err,min :=

1

2(1− �ρ⊗n − (RAρR

†A)

⊗n�1) −→ e−nξ(ρ,RAρRA) =: Q(ρ, RAρR†A)

n

n � 1

worst case with respectto the choice of the initial state (1)

DA→B(ρ) := 1− maxRA∈S

Q(ρ, RAρR†A) Q = min

0≤s≤1Tr[ρs(RAρR

†A)

1−s]

We will only require that : I /∈ S I ∈ S =⇒ DA→B(ρ) = 0

RA

IAIARA. . . ˜ρ

}ρ⊗n

˜ρ ˜ρ(1)

(2)

R��A

R�A

(IAρIA)⊗n (p = 1/2)

(R?AρR

?†A )⊗n (p = 1/2)

(3)

(4)S

A

B

. . .

. . .

RobertAlice


* K. Modi et al, Rev. Mod. Phys. 84, 1665 (2012), F. Ciccarello et al, New J. Phys. 16 , 013038 (2014)

We set

Λ = diag{λ1, . . . ,λdA},λi > λi+1

RA = exp[i(UAΛU†A)]

{ HΛA

We have proved that:

DΛA→B

(ρ) = 1−maxH

ΛA

Q(ρ, eiHΛAρe−iH

ΛA)

1. DS is a bona-fide measure of non-classical correlations *

2. for generic bipartite systems:

2.1 for qubit-qudit systems:

3. DS is invariant under constant shifts of the spectrum:

4. for qubit-qudit systems the maximum over the set of separable states is reached by pQC states

5. there exist simple closed expressions for the set of pure states

LQU

DΛA→B(ρ) = UΛ

A→B(ρ) +O(Λ3)





U!A $!% ! min

K!I$!; K!%: (2)


K!A " VAdiag$!%Vy







A B

(a)

(b)

A B



0 14

12

34

10

12

1

p

loca

l unc

erta

inty


z % ofthe "A




240402-2

BADΛ

A→B(ρ) = f(Λ)UΛA→B(ρ)

DΛA→B(ρ) = DΛ+b

A→B(ρ), b ∈ R

Conclusions

Such correlations can be seen as resources in the context of quantum illumination(local quantum uncertainty, discord of response, interferometric power)

There exist non-classical correlations going beyond the definition of entanglement

Discriminating strength: well-defined measure of discord,easy to compute, with a clear operative meaning

��

BONA FIDE MEASURES OF NON-CLASSICAL CORRELATIONS · BONA FIDE MEASURES OF NON-CLASSICAL...

Documents

Transcript of BONA FIDE MEASURES OF NON-CLASSICAL CORRELATIONS · BONA FIDE MEASURES OF NON-CLASSICAL...