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
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
�S(ρB)− S(B|{ΠA
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
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−iϕHA ,
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 PAof 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 from earth 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 PA(ρAB).
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 · �σA. 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 conundrum requires 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 perform the best
possible joint measurement on the resulting global state ρϕAB
,
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,
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
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 system which 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-
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
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−iϕHA ,
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 PAof 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 from earth 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 PA(ρAB).
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 · �σA. 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 conundrum requires 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 perform the best
possible joint measurement on the resulting global state ρϕAB
,
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,
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
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 system which 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-
e−iϕHA
?
Measures of discord related to q. metrology
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)
D(B|A) = S(ρA)− S(ρAB) + minΠA
j
�S(ρB)− S(B|{ΠA
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
Measures of discord related to q. metrology
(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
The discriminating strength
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
The discriminating strength
* 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)
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
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
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