© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

On the failure of subadditivity of the Wigner-Yanase entropy

Robert Seiringer

Published Jan 1, 2007• 5 citations

Abstract

It was recently shown by Hansen that the Wigner-Yanase entropy is, for general states of quantum systems, not subadditive with respect to decomposition 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 applications 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 S WY (ρ, K) = 1 2 Tr[ρ1/2, K] 2 = Trρ 1/2 Kρ 1/2 K − TrρK 2 for density matrices ρ of quantum systems, with K some fixed self-adjoint operator. They showed that S WY is concave in ρ [4, 5] and, for pure states, subadditive with respect to decomposition of the quantum system into two subsystems.

Related papers

Inequalities for quantum entropy: A review with conditions for equality

Mary Beth Ruskai

This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.

arXiv:quant-ph/0205064•Journal of Mathematical Physics•Sep 1, 2002•185 citations

Strong subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states

Giancarlo Camilo, Gabriel Teixeira Landi, Sebas Eliëns

Recently, there has been a surge of interest in using R\'enyi entropies as quantifiers of correlations in many-body quantum systems. However, it is well known that in general these entropies do not satisfy the strong subadditivity inequality, which is a central property ensuring the positivity of correlation measures. In fact, in many cases they do not even satisfy the weaker condition of subadditivity. In the present paper we shed light on this subject by providing a detailed survey of R\'enyi entropies for bosonic and fermionic Gaussian states. We show that for bosons the R\'enyi entropies always satisfy subadditivity, but not necessarily strong subadditivity. Conversely, for fermions both do not hold in general. We provide the precise intervals of the R\'enyi index $\ensuremath{\alpha}$ for which subadditivity and strong subadditivity are valid in each case.

arXiv:1810.07070•Physical review. B./Physical review. B•Jan 31, 2019•14 citations

On the failure of subadditivity of the Wigner–Yanase entropy

Robert Seiringer

arXiv:0704.1435•Letters in Mathematical Physics•May 30, 2007•9 citations

A Fundamental Property of Quantum-Mechanical Entropy

Élliott H. Lieb, Mary Beth Ruskai

We have proved the strong subadditivity of quantum-mechanical entropy and the Wigner-Yanase-Dyson conjecture.Received 26 December 1972DOI:https://doi.org/10.1103/PhysRevLett.30.434©1973 American Physical Society

Physical Review Letters•Mar 5, 1973•237 citations

The Wigner-Yanase Entropy is not Subadditive

Frank Hansen

arXiv:math-ph/0609019•Journal of Statistical Physics•Jan 17, 2007•30 citations

Quantum weighted entropy and its properties

Yu. M. Suhov, Stefan Zohren

We introduce quantum weighted entropy in analogy to an earlier notion of (classical) weighted entropy and derive many of its properties. These include the subadditivity, concavity and strong subadditivity property of quantum weighted entropy, as well as an analog of the Araki-Lieb inequality. Interesting byproducts of the proofs are a weighted analog of Klein's inequality and non-negativity of quantum weighted relative entropy. A main difficulty is the fact that the weights in general do not commute with the density matrices.

arXiv:1411.0892•Nov 4, 2014•4 citations

It was recently shown by Hansen that the Wigner-Yanase entropy is, for general states of quantum systems, not subadditive with respect to decomposition 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 applications 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 S WY (ρ, K) = 1 2 Tr[ρ1/2, K] 2 = Trρ 1/2 Kρ 1/2 K − TrρK 2 for density matrices ρ of quantum systems, with K some fixed self-adjoint operator. They showed that S WY is concave in ρ [4, 5] and, for pure states, subadditive with respect to decomposition of the quantum system into two subsystems.

This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.

Recently, there has been a surge of interest in using R\'enyi entropies as quantifiers of correlations in many-body quantum systems. However, it is well known that in general these entropies do not satisfy the strong subadditivity inequality, which is a central property ensuring the positivity of correlation measures. In fact, in many cases they do not even satisfy the weaker condition of subadditivity. In the present paper we shed light on this subject by providing a detailed survey of R\'enyi entropies for bosonic and fermionic Gaussian states. We show that for bosons the R\'enyi entropies always satisfy subadditivity, but not necessarily strong subadditivity. Conversely, for fermions both do not hold in general. We provide the precise intervals of the R\'enyi index $\ensuremath{\alpha}$ for which subadditivity and strong subadditivity are valid in each case.

We introduce quantum weighted entropy in analogy to an earlier notion of (classical) weighted entropy and derive many of its properties. These include the subadditivity, concavity and strong subadditivity property of quantum weighted entropy, as well as an analog of the Araki-Lieb inequality. Interesting byproducts of the proofs are a weighted analog of Klein's inequality and non-negativity of quantum weighted relative entropy. A main difficulty is the fact that the weights in general do not commute with the density matrices.