Theoretical and Mathematical Physics, 143(2): 681–688 (2005)
QUANTUM VERSUS CLASSICAL UNCERTAINTY
S. L. Luo∗
The uncertainty of an observable in a quantum state is usually described by variance. This description
is well suited when the states are pure. But when the states are mixed, things become subtle, and the
variance is a hybrid of quantum and classical uncertainties. Motivated by the notion of Fisher information
in statistical inference, we establish a decomposition of the variance into quantum and classical parts. The
key observation is that the Wigner–Yanase skew information (a distinguished version of quantum Fisher
information) can be interpreted as a measure of quantum uncertainty. We also establish a decomposition
of the conventional covariance into quantum and classical parts. The results provide a new perspective for
understanding uncertainty and correlation and are used to quantify entanglement, as well as to establish
a new uncertainty relation in purely quantum terms.
Keywords: uncertainty, classical uncertainty, quantum uncertainty, skew information, quantum correlation, entanglement, uncertainty principle
1. Introduction
Traditionally, from a fundamental standpoint, a quantum state is tacitly assumed to be pure, with the
rationale that mixed states only arise in a noisy environment and can be expressed as convex combinations
of pure states (but not vice versa). For a variety of reasons, this view seriously obscures the subtle and
fundamental characteristics of mixed states.
First, while a mixed state can be expressed as a convex combination of pure states, the representation
is never unique! Consequently, the reduction of mixed states to pure ones causes unexpected difficulties
and thorny problems in studying mixed-state uncertainty and entanglement. In fact, the behaviors of
entanglement and the violations of the Bell inequalities for mixed states are much more intriguing and
elusive than those for pure ones [1].
Second, when a composite quantum system is in a pure state that is entangled, its subsystems are
in mixed states. This is the typical scenario for quantum measurements, as well as for entanglement
quantifications in which the reduced density operators play a crucial role [2].
Third, as recently demonstrated by Henderson et al. [3], not all mixed states can be obtained from pure
ones by local noise contaminations. This greatly restricts the class of mixed states of multipartite systems
arising in noisy environments if the noise is local.
Finally, in classical game theory and statistical inference theory, it is well known that strategies based
on mixed states may be better than any pure ones in general situations, and the famous existence theorem
for the Nash equilibrium requires mixed strategies (based on mixed states) as fundamental ingredients [4].
If we take the position that quantum theory is not so much about nature but more about our statistical
knowledge of and information synthesis on nature (see, e.g., [5], [6], and the references therein) and if we
pursue a statistical inference approach to quantum theory, then we must regard mixed states as a primary
∗ Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, P. R. China; School of Mathematics and Statistics, Carleton University, Ottawa, Canada, e-mail: luosl@amt.ac.cn.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 231–240, May, 2005. Original
article submitted June 29, 2004.
c 2005 Springer Science+Business Media, Inc.
0040-5779/05/1432-0681 681
input. This is also corroborated by the Gleason theorem, which states that in a very general sense, any
quantum state is given by a mixed state [7]. Mermin even stressed that the mixed states and the ensuing
correlations constitute the only fundamental, irreducible, and objective physical reality [8].
The unfortunate (fortunate?) feature of mixed states is that they incorporate both classical mixing
information and quantum state information. These two quite different kinds of information are intertwined,
and their interplay with superposition and interference usually yields complicated and ambiguous results.
The problem of how to disentangle them has both theoretical value and practical importance because of the
recently increasing interest in quantum information theory [9]. Several authors have addressed this issue in
the context of separating correlations into quantum and classical parts [10], and a large body of literature
is devoted to quantifying entanglement as nonlocal quantum correlation [2]. In all these works, the main
tools are various notions related to the von Neumann or relative entropy and to communication theory. In
this paper, we use a quantum analogue of Fisher information to attack this disentangling problem in the
context of uncertainty induced by mixed states.
In the standard practice and formalism of quantum mechanics, the uncertainty of an observable A in
a quantum state ρ is usually quantified by the variance as
V (ρ, A) := tr ρA20 = tr ρA2 − (tr ρA)2 ,
where tr denotes the trace, A0 := A − tr ρA. In particular, when ρ = |ψψ| is pure,
V |ψψ|, A = ψ|A2 |ψ − ψ|A|ψ2 .
The description of uncertainty in terms of variance serves well when the states are pure. But things
become complicated and subtle when the states are mixed because two kinds of contributions, one quantum
and the other classical, are then mixed together in the total variance. It is desirable to split the uncertainty
(variance) into two parts:
Uncertainty = Quantum Part + Classical Part.
Once this is achieved, it can also be used to separate correlations into quantum and classical parts via a
polarization procedure.
Therefore, we seek a decomposition
V (ρ, A) = Q(ρ, A) + C(ρ, A),
where Q(ρ, A) and C(ρ, A) correspond to the respective quantum and classical uncertainties and V (ρ, A)
denotes the total uncertainty (variance). Of course, we cannot expect a unique and universal solution to
such a problem, but for the decomposition to be intuitive and reasonable, the following requirements should
be met:
1. As a measure of quantum uncertainty, Q(ρ, A) should be convex (subadditive) with respect to ρ
because classical mixing does not increase quantum uncertainty. Therefore,
Q
λi ρi , A ≤
λi Q(ρi , A)
i
i
i
i
for i λi = 1, λi ≥ 0, and quantum states ρi . In contrast, as a measure of classical uncertainty,
C(ρ, A) should be concave (superadditive) with respect to ρ because classical mixing increases
classical uncertainty. Therefore,
λi ρi , A ≥
λi C(ρi , A).
C
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2. In the extreme case where ρ is pure, we should have Q(ρ, A) = V (ρ, A) and C(ρ, A) = 0 because
there is no classical mixing and all uncertainties are quantum.
3. When ρ and A commute, we should have Q(ρ, A) = 0 and C(ρ, A) = V (ρ, A) because we can
work in a representation diagonalizing ρ and A simultaneously. In this representation, both ρ
and A behave like classical variables, and all uncertainties are therefore classical. Alternatively,
it can be argued that when ρ and A commute, any eigenstate of ρ is also an eigenstate of A.
But the uncertainty of A in any eigenstate vanishes, and ρ is a classical mixing of its eigenstates.
Therefore, the quantum uncertainty should also vanish because classical mixing does not increase
quantum uncertainty.
With these intuitive requirements in mind and motivated by the notion of Fisher information in statistical
inference [11], [12], we first propose two natural subdivisions of variance for quantum states in Sec. 2. We
then use the new uncertainty measure to study quantum correlations (entanglement) and the Heisenberg
uncertainty principle in Sec. 3. Finally, we conclude with some discussion in Sec. 4.
2. Decomposition of uncertainty: Quantum versus classical
We propose the following two quantities as some uncertainty measures for an observable A in a quantum
state ρ:
P (ρ, A) :=
1
tr{ρ1/2 , A0 }2 ,
2
1
Q(ρ, A) := − tr[ρ1/2 , A0 ]2 ,
2
where {A, B} := AB + BA is the anticommutator, [A, B] := AB − BA is the commutator, and A0 =
A − tr ρA. We note that because the commutator of two self-adjoint operators is skew-adjoint, the minus
sign in the definition of Q(ρ, A) is necessary to guarantee that this quantity is positive. We show that
P (ρ, A) and Q(ρ, A) have significant informational and physical contents. Clearly, Q(ρ, A) can also be
written as
1
Q(ρ, A) = − tr[ρ1/2 , A]2 ,
2
i.e., we need not displace A to have a zero average. But this is not the case for P (ρ, A). Formally, Q(ρ, A)
and P (ρ, A) can also be written as
P (ρ, A) = tr ρA20 + tr ρ1/2 A0 ρ1/2 A0 ,
(1)
Q(ρ, A) = tr ρA20 − tr ρ1/2 A0 ρ1/2 A0 .
(2)
We further introduce
C(ρ, A) :=
1
P (ρ, A) − Q(ρ, A) .
2
Clearly,
C(ρ, A) = tr ρ1/2 A0 ρ1/2 A0 .
(3)
The above quantities have the relations
a. P (ρ, A) = Q(ρ, A) = V (ρ, A) if ρ is pure,
b. P (ρ, A) ≥ V (ρ, A) ≥ Q(ρ, A) for any mixed state,
c. P (ρ, A) + Q(ρ, A) = 2V (ρ, A) for any mixed state, and
d. P (ρ, A) − Q(ρ, A) = 2C(ρ, A) for any mixed state.
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Only relation b needs a proof. By the cyclic property of the trace, we have
tr ρ1/2 A0 ρ1/2 A0 = tr(ρ1/4 A0 ρ1/4 )(ρ1/4 A0 ρ1/4 ),
which is nonnegative because ρ1/4 A0 ρ1/4 is self-adjoint. Consequently, by Eq. (2),
Q(ρ, A) ≤ tr ρA20 = V (ρ, A).
The other inequality follows similarly.
We now have three measures of uncertainty: V (ρ, A), P (ρ, A), and Q(ρ, A). They coincide for pure
states but differ for mixed ones. Relative to the conventional variance V (ρ, A), the latter two stand in a
symmetric place. It is appealing and convenient to focus on V (ρ, A) and Q(ρ, A) because the former is
familiar and ubiquitous, while the latter involves the commutator and is moreover actually an old concept
with intrinsic informational meaning, although most physicists ignore it: Q(ρ, A) is precisely the skew
information introduced by Wigner and Yanase in 1963, when they studied the information contents of
quantum states [13]. Their interpretation is that Q(ρ, A) quantifies the information contents of the quantum
state ρ with respect to observables not commuting with (i.e., skew to) the observable A, which serves as
a conserved quantity. This interpretation has its root in both quantum measurement theory [14] and
statistical inference theory as a kind of quantum Fisher information [12].
Our key observation now is that Q(ρ, A) can also be regarded as quantifying the information of observables not commuting with (skew to, conjugate to) A in the state ρ. Accordingly, because of the Bohr
complementarity principle, we can further interpret Q(ρ, A) as some kind of uncertainty of A itself in ρ.
Indeed, Q(ρ, A) is just the variance V (ρ, A) when ρ is a pure state.
From Eqs. (2) and (3), we have the following decomposition of the conventional variance, which is our
central concern:
V (ρ, A) = Q(ρ, A) + C(ρ, A).
(4)
We interpret Q(ρ, A) as the quantum part and C(ρ, A) as the classical part of the total uncertainty (variance) V (ρ, A). By direct verifications and the results of Wigner–Yanase [13] and Lieb [15], we see that
this decomposition satisfies our postulated requirements 1–3. Furthermore, Q(ρ, A) and C(ρ, A) have the
following remarkable properties [12], [14], [15]:
1. For any unitary operator U ,
Q(U ρU † , U AU † ) = Q(ρ, A).
In particular, Q(U ρU † , A) = Q(ρ, A) if U commutes with A. Particularly interesting is U = e−itA ,
t ∈ R, for which the state ρ evolves into ρt = e−itA ρeitA according to the Landau–von Neumann
equation. In this scenario, Q(ρt , A) = Q(ρ, A), which means that quantum uncertainty remains
constant for an isolated system with Landau–von Neumann dynamics.
2. Let ρ1 and ρ2 be two quantum states describing the respective first and the second systems, and
let A1 and A2 be two observables. Then
Q(ρ1 ⊗ ρ2 , A1 ⊗ I2 + I1 ⊗ A2 ) = Q(ρ1 , A1 ) + Q(ρ2 , A2 ),
where I1 and I2 are the respective identity operators for the first and second systems.
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3. Let ρ be a quantum state on a composite quantum system, and let ρ1 = tr2 ρ be the marginal
quantum state obtained by tracing ρ over the second subsystem. Let A1 be an observable for the
first quantum subsystem. Then
Q(ρ, A1 ⊗ I2 ) ≥ Q(ρ1 , A1 ).
All these relations hold for C(ρ, A) with the inequality reversed.
Therefore, if we use the two-dimensional vector Q(ρ, A), C(ρ, A) to quantify the uncertainty of A in
the state ρ, then we have a more transparent picture showing the different contributions of the quantum
and classical uncertainties to the total uncertainty (variance).
We illustrate Eq. (4) with a typical example. We consider a composite quantum system H1 ⊗ H2
with dim H1 = dim H2 = N . Let H1 and H2 respectively be observables on H1 and H2 with the spectral
decompositions
H1 =
αi |i1 i1 |,
H2 =
βi |i2 i2 |.
(5)
i
i
Without loss of generality, we assume that |i1 and |i2 constitute respective orthonormal bases for
H1 and H2 . We set A = H1 ⊗ H2 and take two quantum states of the composite system as
ρ = |ΨΨ|,
σ=
1 |ii|,
N
1 |i1 ⊗ |i2 ;
|Ψ = √
N i
(6)
|i = |i1 ⊗ |i2 .
(7)
i
From the correlation standpoint, the states ρ and σ are very different: the former is a Bell-type state that is
maximally entangled, and the latter is simply a classical mixture of product states. The state ρ is pure, all
uncertainties are irreducibly quantum, and we should therefore expect Q(ρ, A) = V (ρ, A) and C(ρ, A) = 0.
The state σ is a classical mixture of eigenstates of A. Because the uncertainty of A in any eigenstate
is zero and classical mixing cannot create quantum uncertainty, we should expect Q(σ, A) = 0. On the
other hand, the classical mixing accounts for all the uncertainties of A in σ, and we should therefore have
C(σ, A) = V (σ, A). Direct evaluation according to Eqs. (2) and (3) indeed yields
2
1 2 2
1 V (ρ, A) =
αi βi −
αi βi ,
N i
N
i
Q(ρ, A) =
2
1 1 2 2
αi βi −
αi βi ,
N i
N
i
C(ρ, A) = 0
and
2
1 2 2
1 V (σ, A) =
α β −
αi βi ,
N i i i
N
i
Q(σ, A) = 0,
2
1 2 2
1 C(σ, A) =
α β −
αi βi ,
N i i i
N
i
which are exactly as expected. Therefore, we see that the quantum uncertainty can distinguish ρ from σ,
but because V (ρ, A) = V (σ, A), the conventional variance cannot distinguish between them.
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3. Quantum correlations and uncertainty relations
Uncertainties are intimately related to correlations. Simple formal generalizations of the quantum and
classical uncertainties lead to the corresponding measures for quantum and classical correlations. Let ρ be
a quantum state, and let A and B be any two operators (not necessarily observables). In contrast to the
conventional covariance
Covρ (A, B) := tr ρA†0 B0
(where A0 = A − tr ρA, B0 = B − tr ρB, and † denotes the adjoint) and motivated by Eqs. (2) and (3), we
define two new measures of correlations by formally polarizing the quantum uncertainty Q(ρ, A) and the
classical uncertainty C(ρ, A):
Qρ (A, B) := tr ρA†0 B0 − tr ρ1/2 A†0 ρ1/2 B0 ,
Cρ (A, B) := tr ρ1/2 A†0 ρ1/2 B0 .
Accordingly, we have the decomposition of the conventional covariance as
Covρ (A, B) = Qρ (A, B) + Cρ (A, B).
Here, we call Qρ (A, B) the quantum correlation because it accounts for the quantum part of the covariance in
a certain sense, and we call Cρ (A, B) the classical correlation because it accounts for the classical part of the
covariance. The quantum correlation Qρ (A, B) is a powerful tool for detecting quantum correlations because
it is a purely quantum measure stripped of classical correlation; moreover, there is the freedom to choose the
measured (tested) observables A and B. In particular, when A and B commute, the quantum correlation
Qρ (A, B) has the following interpretation as a formal generalization of the cross Fisher information for
joint probability densities. Let ρs,t satisfy the Landau–von Neumann equations with both the parameters
s and t,
i
∂ρs,t
= [A, ρs,t ],
∂s
i
∂ρs,t
= [B, ρs,t ],
∂t
ρ0,0 = ρ.
Then
√
√
∂ ρs,t ∂ ρs,t
1
Qρ (A, B) = − tr
,
2
∂s
∂t
which is reminiscent of the classical cross Fisher information. To prove the assertion, we note that A and
B commute and therefore Qρ (A, B) = Qρ (B, A). Moreover, the above Landau–von Neumann equations
imply that
ρs,t = e−isA−itB ρeisA+itB ,
s, t ∈ R,
and hence
√
∂ ρs,t
= ie−isA−itB [ρ1/2 , A]eisA+itB ,
∂s
√
∂ ρs,t
= ie−isA−itB [ρ1/2 , B]eisA+itB .
∂t
The results follow readily by noting the cyclic property of trace.
The quantum correlation has the properties
a. Qρ (A, B) = Qρ (B, A),
b. Qρ (A − a, B − b) = Qρ (A, B) for any real constants a and b,
c. Qρ (aA, bB) = abQρ (A, B) for any real constants a and b, and
d. QUρU −1 (A, B) = Qρ (U −1 AU, U −1 BU ) for any unitary operator U .
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How can our measure Qρ (A, B) of quantum correlation be used to characterize entanglement? To
address this problem, we stress that entanglement is a nonlocal quantum correlation. If we use the quantum
correlation Qρ (A, B) to quantify entanglement, then we should choose A and B as local observables in
different component systems. Then Qρ (A, B) is the quantum correlation of the state ρ, tested on the two
local observables A and B. At this point, it is natural to introduce a new measure of entanglement:
E(ρ) := sup Qρ (A, B),
A,B
where the supremum is taken over all A and B of the forms A = H1 ⊗ I2 and B = I1 ⊗ H2 , H1 and H2 are
respective observables with norms less than unity on H1 and H2 , and ρ is a quantum state in the composite
quantum system H1 ⊗ H2 .
We now consider an example. Let H1 and H2 be as in Eq. (5), and let the states ρ and σ be given by
Eqs. (6) and (7). Let
A = H1 ⊗ I2 ,
B = I1 ⊗ H2 .
Direct calculation then leads to
Covρ (A, B) = Covσ (A, B) =
Qρ (A, B) =
1 (αi − ᾱ)(βi − β̄),
N i
1 (αi − ᾱ)(βi − β̄),
N i
Cρ (A, B) = 0,
Cσ (A, B) =
Qσ (A, B) = 0,
1 (αi − ᾱ)(βi − β̄),
N i
where ᾱ = (1/N ) i αi and β̄ = (1/N ) i βi . Thus, we see that the quantum correlation (and hence also
the classical correlation) distinguishes the states ρ and σ while the conventional covariance fails to do so.
Moreover, from a simple analysis, we see that E(σ) = 0 for all N , E(ρ) = 1 for even N , and E(ρ) = 1 − 1/N 2
for odd N .
As a further application of the measures of quantum uncertainty and quantum correlation, we have
the uncertainty relation
2
Q(ρ, A)Q(ρ, B) ≥ Re Qρ (A, B) ,
which is expressed in purely quantum terms. Here, Re denotes the real part of a complex number. The
proof follows because Re Qρ ( · , · ) is an inner product to which the Schwarz inequality can be applied.
4. Conclusions
In summary, we have proposed some new measures for uncertainties and correlations using the Wigner–
Yanase skew information. This leads to decompositions splitting the conventional variance and covariance
into quantum and classical parts and further provides a new tool for investigating the relations between
the quantum and classical aspects of mixed states and correlations. The approach is in the general vein of
understanding quantum mechanics from an informational perspective [6], [16]–[19].
Acknowledgments. The author is very grateful to Professors Miklos Csorgo and Barbara Szyszkowicz
for the invitation.
This work is supported by an NSERC Canada Grant at Carleton University.
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