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arXiv:quant-ph/0105072v3 31 Oct 2001

Quantum discord: A measure of the quantumness of correlations

Harold Ollivier and Wojciech H. Zurek

Theoretical Division T-6, MS B288, LANL, Los Alamos, NM 87545.

Two classically identical expressions for the mutual information generally diﬀer when the systems

involved are quantum. This diﬀerence deﬁnes the quantum discord. It can be used as a measure of

the quantumness of correlations. Separability of the density matrix describing a pair of systems does

not guarantee vanishing of the discord, thus showing that absence of entanglement does not imply

classicality. We relate this to the quantum superposition principle, and consider the vanishing of

discord as a criterion for the preferred eﬀectively classical states of a system, i.e. the pointer states.

PACS numbers: 03.65.Ta, 03.65.Yz, 03.67.-a

The original motivation for the pointer states — states

that are monitored by the environment but do not entan-

gle with it, and are therefore stable in spite of the open-

ness of the system — comes from the study of quantum

measurements [1, 2, 3]. When the quantum apparatus A

interacts with the system S(pre-measurement), the S–

Apair becomes entangled. The nature of the resulting

quantum correlations makes it impossible to ascribe any

independent reality to, say, the state of the apparatus [4].

For, a measurement of diﬀerent observables on the state

of the system will force the apparatus into mutually in-

compatible pure quantum states. This is a consequence

of the basis ambiguity. It is best exhibited by noting that

the S–Astate after the pre-measurement,

|ψS,AiP=X

i

αi|sii|aii(1)

is typically entangled. One can rewrite it in a diﬀer-

ent basis of, e.g. the system, and one-to-one correlation

with a corresponding set of pure, but not necessarily or-

thogonal, states of the apparatus will remain. Thus, it

is obviously impossible to maintain that before the mea-

surements the apparatus had a an unknown but real (i.e.,

existing independently of the system) quantum state.

Decoherence leads to environment-induced superselec-

tion (einselection) which singles out the pointer states

and thus removes quantum excess of correlation respon-

sible for the basis ambiguity. The density matrix of

the decohering quantum apparatus loses its oﬀ-diagonal

terms as a result of the interaction with the environment

[5, 6, 7, 8]:

ρP

S,A=|ψS,AiPhψS,A|P

→X

i|αi|2|siihsi| ⊗ |aiihai|=ρD

S,A.(2)

Above hai|aji=δi,j following the ideal einselection pro-

cess, which transforms a pure ρP

S,Ainto a decohered ρD

S,A

satisfying the superselection identity [9, 10]:

ρD

S,A=X

i

PA

iρD

S,APA

i.(3)

Above PA

icorrespond to the superselection sectors of the

apparatus, e.g. the record states of its pointer (in our ex-

ample PA

i=|aiihai|). One implication of this equation

is that — once einselection has forced the apparatus to

abide by Eq. (3) — its state can be consulted (measured)

in the basis corresponding to the superselection sectors

PA

ileaving ρD

S,Aunchanged [5, 8].

Einselection, Eq. (2), obviously decreases correlations

between Sand A. Yet, in a good measurement, one-

to-one correlations between the pointer states of the ap-

paratus and a corresponding set of system states must

survive. We shall use two classically equivalent formu-

lae for the mutual information to quantify the quantum

and the classical strength of the correlations present in

a joint density matrix ρS,A, and study the diﬀerence be-

tween these two as a measure of the quantum excess of

correlations — the quantum discord — in ρS,A.

Mutual Information — In classical information theory

[11] the entropy, H(X), describes the ignorance about a

random variable X,H(X) = −PxpX=xLog pX=x. The

correlation between two random variables Xand Yis

measured by the mutual information:

J(X:Y) = H(X)−H(X|Y),(4)

where H(X|Y) = PypY=yH(X|Y =y) is the conditional

entropy of Xgiven Y. All the probability distributions

are derived from the joint one, pX,Y:

pX=X

y

pX,Y=y, pY=X

x

pX=x,Y(5)

pX |Y=y=pX,Y=y/pY=y(Bayes rule) (6)

Hence, the mutual information measures the average de-

crease of entropy on Xwhen Yis found out. Using

the Bayes rule, Eq. (6), one can show that H(X|Y) =

H(X,Y)−H(Y). This leads to another classically equiv-

alent expression for the mutual information:

I(X:Y) = H(X) + H(Y)−H(X,Y).(7)

One would like to generalize the concept of mutual in-

formation to quantum systems. One route to this goal,

motivated by discussions of quantum information pro-

cessing, has been put forward [12, 13]. We shall pursue

2

a diﬀerent strategy, using Eqs. (4) and (7). We start by

deﬁning Iand Jfor a pair of quantum systems.

I—All the ingredients involved in the deﬁnition of I

are easily generalized to deal with arbitrary quantum sys-

tems by replacing the classical probability distributions

by the appropriate density matrices ρS,ρAand ρS,Aand

the Shannon entropy by the von Neumann entropy, e.g.

H(S) = H(ρS) = −TrSρSLog ρS:

I(S:A) = H(S) + H(A)−H(S,A).(8)

In this formula, H(S) + H(A) represents the uncer-

tainty of Sand Atreated separately, and H(S,A) is

the uncertainty about the combined system described

by ρS,A. However, in contrast with the classical case,

extracting all information potentially present in a com-

bined quantum system described by ρS,Awill, in general,

require a measurement on the combined Hilbert space

HS,A=HS⊗ HA. The quantum version of Ihas been

used some years ago to study entanglement [14], and sub-

sequently rediscovered [15].

J—The generalization of this expression is not as

automatic as for I, since the conditional entropy H(S|A)

requires us to specify the state of Sgiven the state of A.

Such statement in quantum theory is ambiguous until

the to-be-measured set of states Ais selected. We focus

on perfect measurements of Adeﬁned by a set of one-

dimensional projectors {ΠA

j}. The label jdistinguishes

diﬀerent outcomes of this measurement.

The state of S, after the outcome corresponding to ΠA

j

has been detected, is

ρS|ΠA

j= ΠA

jρS,AΠA

j/TrS,AΠA

jρS,A,(9)

with probability pj= TrS,AΠA

jρS,A.H(ρS|ΠA

j) is the

missing information about S. The entropies H(ρS|ΠA

j),

weighted by probabilities, pj, yield to the conditional en-

tropy of Sgiven the complete measurement {ΠA

j}on A,

H(S|{ΠA

j}) = X

j

pjH(ρS|ΠA

j).(10)

This leads to the following quantum generalization of J:

J(S:A){ΠA

j}=H(S)−H(S|{ΠA

j}).(11)

This quantity represents the information gained about

the system Sas a result of the measurement {ΠA

j}.

Quantum discord — The two classically identical ex-

pressions for the mutual information, Eqs. (4) and (7),

diﬀer in a quantum case [16]. The quantum discord is

this diﬀerence,

δ(S:A){ΠA

j}=I(S:A)− J (S:A){ΠA

j}(12)

=H(A)−H(S,A) + H(S|{ΠA

j}).(13)

It depends both on ρS,Aand on the projectors {ΠA

j}.

The quantum discord is asymmetric under the change

S ↔ A since the deﬁnition of the conditional entropy

H(S|{ΠA

j}) involves a measurement on one end (in our

case the apparatus A), that allows the observer to infer

the state of S. This typically involves an increase of

entropy. Hence H(S|{ΠA

j})≥H(S,A)−H(A), which

implies that for any measurement {ΠA

j},

δ(S:A){ΠA

j}≥0.(14)

The proofs are postponed to the end of this letter.

We shall be usually concerned about the set {ΠA

j}that

minimizes the discord given a certain ρS,A. Minimizing

the discord over the possible measurements on Acorre-

sponds to ﬁnding the measurement that disturbs least

the overall quantum state and that, at the same time, al-

lows one to extract the most information about S. Deco-

herence picks out a set of stable states and converts their

possible superpositions into mixtures, Eq. (2). Moreover,

an unread measurement {ΠA

j}on the apparatus has an

eﬀect analogous to einselection in the corresponding ba-

sis through the reduction postulate [1]. Hence it is rather

natural to expect that when the set {ΠA

j}corresponds to

the superselection sectors {PA

i}of Eq. (3), there would

be no extra increase of entropy:

ρS,A=X

j

ΠA

jρS,AΠA

j⇒δ(S:A){ΠA

j}= 0.(15)

Thus, following einselection, the information can be ex-

tracted from S–Awith a local measurement on Awith-

out disturbing the overall state. The state of Scan be

inferred from the outcome of the measurement on Aonly.

The converse of Eq. (15) is also true:

δ(S:A){ΠA

j}= 0 ⇒ρS,A=X

j

ΠA

jρS,AΠA

j.(16)

Hence, a vanishing discord can be considered as an in-

dicator of the superselection rule, or — in the case of

interest — its value is a measure of the eﬃciency of eins-

election. When δis large for any set of projectors {ΠA

j},

a lot of information is missed and destroyed by any mea-

surement on the apparatus alone, but when δis small

almost all the information about Sthat exists in the S–

Acorrelations is locally recoverable from the state of the

apparatus.

The quantum discord can be illustrated in a simple

model of measurement. Let us assume the initial state

of Sis (|0i+|1i)/√2. The pre-measurement is a c-not

gate yielding |ψS,AiP= (|00i+|11i)/√2. If |0iand |1i

of Aare pointer states, partial decoherence will suppress

oﬀ-diagonal terms of the density matrix:

ρS,A=1

2(|00ih00|+|11ih11|)

+z

2(|00ih11|+|11ih00|),(17)

3

with 0 ≤z < 1. Fig. 1 shows δfor various values of z

and various bases of measurement parametrized by θ,

{cos(θ)|0i+eiφ sin θ|1i, e−iφ sin θ|0i − cos θ|1i},(18)

with φ= 1rad. Only in the case of complete einselec-

tion (z= 0) there exist a basis for which discord dis-

appears. The corresponding basis of measurement is

{|0i,|1i} (θ= 0), i.e. it must be carried out in the

pointer basis.

FIG. 1: Discord for the states given in Eq. (17), with the

measurement basis deﬁned as in Eq. (18).

Classical aspect of quantum correlations — Separabil-

ity has been often regarded as synonymous of classicality.

The temptation that leads one to this conclusion starts

with an observation that — by deﬁnition — a separable

density matrix is a mixture of density matrices

ρS,A=X

i

piρi

S,A(19)

that have explicit product eigenstates,

ρi

S,A=X

j

p(i)

j

s(i)

jE

a(i)

jEDa(i)

j

Ds(i)

j

(20)

and hence classical correlations. One might have thought

that mixing such obviously classical density matrices can-

not bring in anything quantum: After all, it involves only

loss of information — forgetting of the label iin ρi

S,A.

Yet this is not the case. One symptom of the quantum-

ness of a separable ρS,Awith non-zero discord is imme-

diately apparent: Unless there exists a complete set of

projectors {ΠA

j}for which δ(S:A){ΠA

j}= 0, ρS,Ais

perturbed by all local measurements. By contrast, when

δ(S:A){ΠA

j}= 0, then the measurement {ΠA

j}on A,

and an appropriate conditional measurement (i.e. con-

ditioned by the outcome of the measurement on A) will

reveal all of the information in S–A, i.e. the resulting

state of the pair will be pure. Moreover this procedure

can be accomplished without perturbing the ρS,Afor an-

other observer, a bystande! r not aware of the outcomes.

Thus, for each outcome jthere exist a set {πS

j,k}of

conditional one dimensional projectors such that

ρS,A=X

jX

k

πS

j,kΠA

jρS,AΠA

jπS

j,k ,(21)

and πS

j,kΠA

jρS,AΠA

jπS

j,k is pure for any jand k. Above,

the sets {πS

j,k}for diﬀerent jwill not coincide in gen-

eral ({πS

j,k}is a function of j) and do not need to com-

mute. Classical information is locally accessible, and can

be obtained without perturbing the state of the system:

One can interrogate just one part of a composite system

and discover its state while leaving the overall density

matrix (as perceived by observers that do not have ac-

cess to the measurement outcome) unaltered. A general

separable ρS,Adoes not allow for such insensitivity to

measurements: Information can be extracted from the

apparatus but only at a price of perturbing ρS,A, even

when this density matrix is separable. However, when

discord disappears, such insensitivity (which may be the

deﬁning feature of “classical reality”, as it allowes ac-

quisition of information without perturbation of the un-

derlying state) becomes possible for correlated quantum

systems. This quantum character of separable density

matrices with non zero discord is a consequence of the

superposition principle for A, since more than one basis

n

a(i)

jEojfor the apparatus is needed in Eq. (20) in order

to warrant a non vanishing discord.

The diﬀerence between separability and vanishing dis-

cord can be illustrated by a speciﬁc example. Fig. 2

shows discord for a Werner state ρS,A=1−z

41+z|ψihψ|

with |ψi= (|00i+|11i)/√2. It can be seen that dis-

cord is greater than 0 in any basis when z > 0, which

contrasts with the well-known separability of such states

when z < 1/3.

Conclusion — The quantum discord is a measure of

the information that cannot be extracted by the reading

of the state of the apparatus (i.e. without joint mea-

surements). Hence the quantum discord is a good in-

dicator of the quantum nature of the correlations. The

pointer states obtained by minimizing the quantum dis-

cord over the possible measurements should coincide with

the ones obtained with the predictability sieve criterion

[5, 7], hence showing that the accessible information re-

mains in the most stable pointer states.

—

Proposition 1. H(S|{ΠA

j}) = H(ρD

S,A)−H(ρD

A), with

ρD

S,A=PjΠA

jρS,AΠA

j.

Proof 1. ρD

S,Ais block-diagonal. The j-th block equals

pjρS|ΠA

j. By doing calculations block by block one has:

4

Non−separableSeparable

z

δ

FIG. 2: Value of the discord for Werner states 1−z

41+z|ψihψ|,

with |ψi= (|00i+|11i)/√2. Discord does not depend on the

basis of measurement in this case because both 1and |ψiare

invariant under rotations.

H(ρD

S,A) = PjH(pjρS|ΠA

j) = PjpjH(ρS|ΠA

j)−

PjpjLog pj=H(S|{ΠA

j})−H(ρD

A),which completes

the proof.

Proposition 2. δ(S:A){ΠA

j}≥0.

Proof 2. This is a direct consequence of the previous

proposition and the concavity of H(S,A)−H(A) [17].

Proposition 3. δ(S:A){ΠA

j}= 0 ⇔ρS,A=

PjΠA

jρS,AΠA

j.

Proof 3. Proposition 1 already shows the converse.

To prove the direct implication we will start with ρS,A

and {ΠA

j}, a complete set of orthogonal projectors, such

that δ(S:A){ΠA

j}= 0. Without loss of generality,

we can write the density matrix of S–Aas ρS,A=

PjΠA

jρS,AΠA

j+ additional terms.If we choose {|sii} a

basis of S, and {

ak|j}ka basis of the subspace of Ade-

ﬁned by ΠA

j, the general form of the additional terms in

the above formula will be c(si, si′, ak|j, ak′|j′)|siihsi′| ⊗

ak|jak′|j′

with j6=j′. Suppose that one of those co-

eﬃcients is non-zero. By changing the basis {|sii}i, we

can suppose i6=i′. We introduce now a new density ma-

trix ˆρS,Aobtained from ρS,Aby removing the preceding

matrix element and its complex conjugate. This ensures

that ˆρS,Ais associated with a physical state. This state

satisﬁes H(ˆρS,A)> H(ρS,A) and H( ˆρA) = H(ρA).The

concavity of H(S,A)−H(A) implies inequalities;

H(ρS,A)−H(ρA)< H(ˆρS,A)−H(ˆρA),

H(ˆρS,A)−H(ˆρA)≤H(ρD

S,A)−H(ρD

A).

Then δ(S:A){ΠA

j}<0, which contradicts our primary

assumption and proves our last result.

Remark. We deﬁned Jwith the help of a measure-

ment associated with one-dimensional pro jectors. One

can be interested in looking at multi-dimensional pro-

jective measurements. Depending on the context, two

diﬀerent generalization can be used.

For measurement purposes, one may adopt

ρS|ΠA

j= TrAΠA

jρS,A/TrS,AΠA

jρS,A,

since all the correlations (quantum as well as classical)

between Sand the subspace of the apparatus deﬁned by

ΠA

jare not observed. Proposition 1 no longer holds, but

using the same techniques we still have δ(S:A){ΠA

j}≥0

and if δ(S:A){ΠA

j}= 0, then ρS,A=PjΠA

jρS,AΠA

j.

For decoherence purposes, one may prefer to deﬁne J

as H(S) + H(A)D−H(S,A)D. With this deﬁnition,

Proposition 3 is valid. Jnow represents the average in-

formation, quantum and classical that remains in the pair

S–Aafter a decoherence process leading to einselection

of the superselection sectors {ΠA

j}.

This research was supported in part by NSA. Pre-

liminary results were presented at the Newton Institute

in July 1999 by W.H.Z., at the Ushuaia PASI in Oc-

tober 2000 by H.O., and described in the proceedings

of the 100th anniversary of Planck’s Constant Meeting

[16]. Useful conversations with E. Knill, R. Laﬂamme, B.

Schumacher and L. Viola are gratefully acknowledged.

Note — After completion of this work, we became

aware of a related work by L. Henderson and V. Ve-

dral (quant-ph/0105028).

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