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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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