On the failure of subadditivity of the Wigner-Yanase entropy
On the failure of subadditivity of the
Wigner-Yanase entropy
Robert Seiringer
Department of Physics, Jadwin Hall, Princeton University
P.O. Box 708, Princeton, NJ 08542, USA
Email: rseiring@math.princeton.edu
April 3, 2007
Abstract
It was recently shown by Hansen that the Wigner-Yanase entropy is,
for general states of quantum systems, not subadditive with respect to de-
composition into two subsystems, although this property is known to hold
for pure states. We investigate the question whether the weaker property
of subadditivity for pure states with respect to decomposition into more
than two subsystems holds. This property would have interesting appli-
cations in quantum chemistry. We show, however, that it does not hold
in general, and provide a counterexample.
In 1963, Wigner and Yanase [4] introduced the entropy-like quantity
SWY(ρ,K) = 1
Tr [ρ1/2,K]2 = Tr ρ1/2Kρ1/2K − Tr ρK2 (1)
for density matrices ρ of quantum systems, with K some fixed self-adjoint op-
erator. They showed that SWY is concave in ρ [4, 5] and, for pure states,
subadditive with respect to decomposition of the quantum system into two sub-
systems. More precisely, if |ψ〉 is a normalized vector in the tensor product of
two Hilbert spaces, H1 ⊗H2, and K1 and K2 are self-adjoint operators on H1
and H2, respectively, then
WY(|ψ〉〈ψ|,K1 ⊗ 1+ 1⊗K2) ≤ SWY(ρ1,K1) + SWY(ρ2,K2) , (2)
where ρ1 = TrH2 |ψ〉〈ψ| and ρ2 = TrH1 |ψ〉〈ψ| denote the reduced states of the
subsystems. Recently, it was shown by Hansen [2] that this subadditivity fails
for general mixed states.
Work partially supported by U.S. National Science Foundation grant PHY-0353181 and
by an Alfred P. Sloan Fellowship.
c© 2007 by the author. This paper may be reproduced, in its entirety, for non-commercial
purposes.
http://arxiv.org/abs/0704.1435v1
This leaves open the question whether the Wigner-Yanase entropy is sub-
additive for pure states with respect to decompositions into more than 2 sub-
systems. If true, this property would have interesting consequences concerning
density matrix functionals used in quantum chemistry, as will be explained be-
low. We shall show, however, that this property does not hold, in general.
Let ρ = |ψ〉〈ψ| be a pure state on a tensor product of N Hilbert spaces,
i=1 Hi, and let Ki be self-adjoint operators on Hi. For simplicity we
use the same symbol for the operators on H which act as the identity on the
remaining factors. Subadditivity of SWY would mean that
−SWY(|ψ〉〈ψ|,
iKi) =
− 〈ψ |
iKi|ψ〉
TrHiρiK
i − TrHiρ
i Kiρ
, (3)
where ρi is the reduced density matrix of |ψ〉〈ψ| on Hi.
Assume now that all the Hi are equal to the same H1, say, and that also all
the Ki are equal, i.e., Ki acts as K on the i’th factor for some fixed operator
K on H1. Ineq. (3) together with concavity of SWY would thus imply that
− 〈ψ |
iKi|ψ〉
2 ≥ TrH1γK2 − TrH1γ1/2Kγ1/2K , (4)
i6=j KiKj
≥ (TrH1γK)
2 − TrH1γ1/2Kγ1/2K , (5)
where γ =
i ρi denotes the one-particle density matrix of |ψ〉〈ψ|. This repre-
sents a correlation inequality, bounding from below two-particle terms in terms
of one-particle terms only.
As explained in [1], the validity of (4) for continuous quantum systems in the
case whereK is the characteristic function of a ball of arbitrary size and location
would imply that the ground state energies of Coulomb systems like atoms
and molecules could be bounded from below by a density-matrix functional
introduced by Müller [3]. For N = 2 this follows from the result in [4].
In the following, we shall show that, in general, (4) fails to hold for N = 3,
and hence for all N ≥ 3. We choose the simplest nontrivial three-particle Hilbert
space, C2 ⊗C2 ⊗C2, and pick a basis {| ↑〉, | ↓〉} in C2. We choose K = | ↑〉〈↑ |,
ψ(↑, ↑, ↑) = 2√
ψ(↑, ↑, ↓) = ψ(↑, ↓, ↑) = ψ(↓, ↑, ↑) = 4√
ψ(↑, ↓, ↓) = ψ(↓, ↑, ↓) = ψ(↓, ↓, ↑) = 1√
ψ(↓, ↓, ↓) = 0 . (6)
1This particular counterexample was found with the aid of the computer algebra software
Mathematica.
〈ψ|ψ〉 = 1
22 + 3 ∗ 42 + 3 ∗ 1
iKi|ψ〉 =
3 ∗ 22 + 2 ∗ 3 ∗ 42 + 1 ∗ 3 ∗ 1
32 ∗ 22 + 22 ∗ 3 ∗ 42 + 1 ∗ 3 ∗ 1
and hence the left side of Ineq. (4) equals
≈ 0.126942 . (8)
The one-particle density matrix γ is given by the 2× 2-matrix
37 16
16 18
whose square root equals
1/2 ≈
5.85827 1.63729
1.63729 3.91399
. (10)
Hence the right side of (4) is
37− (5.85827)2
≈ 0.146221 > 0.126942 . (11)
This shows that Ineq. (4) fails in general for N > 2, and hence the Wigner-
Yanase entropy is not subadditive with respect to the decomposition of pure
states into more than 2 subsystems.
We note that the same counterexample can also be constructed for contin-
uous quantum systems, where K equals the characteristic functions of some
measurable set B. One simply takes B and Ω to be two disjoint sets, each with
volume one, and sets
ψ(x1, x2, x3) =
2 if all 3 particles are in B
4 if 2 particles are in B and 1 in Ω
1 if 1 particle is in B and 2 in Ω
0 otherwise.
This leads to the same counterexample as above.
Similarly, one can construct a counterexample for fermionic (i.e., antisym-
metric) wavefunctions which, after all, is the case of interest in [1]. Simply take
(x, y) as the coordinates of one particle, choose the wave function to be the
product of (12) for the x variables and a Slater-determinant for the y variables,
which is non-zero only if all the y’s are in some set Λ. IfK denotes multiplication
by the characteristic function of B × Λ, this leads to the same counterexample
as before.
References
[1] R.L. Frank, E.H. Lieb, R. Seiringer, H. Siedentop, Müller’s exchange-
correlation energy in density-matrix-functional theory, in preparation
[2] F. Hansen, The Wigner-Yanase entropy is not subadditive, J. Stat. Phys.
126, 643–648 (2007).
[3] A.M.K. Müller, Explicit approximate relation between reduced two- and one-
particle density matrices, Phys. Lett. A 105, 446–452 (1984).
[4] E.P. Wigner, M.M. Yanase, Information content of distributions, Proc. Nat.
Acad. Sci. USA 49, 910–918 (1963).
[5] E.P. Wigner, M.M. Yanase, On the positive semidefinite nature of a certain
matrix expression, Canad. J. Math. 16, 397–406 (1964).
