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数理解析研究所講究録

STRONG SUBADDITIVITY PROPERTY OF ENTROPY IN FERMION SYSTEMS(Recent Trends in Infinite Dimensional Non-Commutative Analysis)

Hajime Moriya

Published Apr 1, 1998• 0 citations
OpenAlex

Abstract

We prove that strong subadditibity(SSA) of entropy holds for fermion systems. Main resultProbably,strong subadditibity(SSA) is the one of the most powerful prop- erties of entropy.In this paper, we prove SSA holds for fermion systems.We consider Fermonic systems on an integer lattice $\mathrm{Z}^{d},\mathrm{t}\mathrm{h}\mathrm{e}$ set of $\mathrm{d}$ -tuples of integers.The creation and annihilation operators will be denoted by $a_{j}^{*}$ and $a_{j}(j\in \mathrm{Z}^{d})$ .They satisfy the canonical anticommutation relations,

Related papers

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

BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

Mary Beth Ruskai

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

Reviews in Mathematical Physics•Jan 1, 1994•161 citations

Addendum Entropy and information causality in general probabilistic theories

Howard Barnum, Jonathan Barrett, Lisa Orloff Clark +5 more

In this addendum to our paper (2010 New J. Phys. 12 033024), we point out that an elementary consequence of the strong subadditivity inequality allows us to strengthen one of the main conclusions of that paper. In this addendum to our paper [1], we point out that an elementary consequence of the strong subadditivity inequality allows us to strengthen one of the main conclusions of that paper. In [1], we defined notions of measurement and mixing entropy for a broad class of probabilistic theories (which includes finite-dimensional quantum theory). We defined a theory to be monoentropic if for every state of a system in the theory, the mixing entropy (denoted S) is equal to the measurement entropy (denoted H ). This broad class of theories has a notion of composite system, and of the ‘marginal state’ of a subsystem given a state for the composite system, so that the strong subadditivity inequality may be formulated for real-valued functions of states, including the measurement and mixing entropies. We proved the following theorem. Theorem 1 (theorem 4 of [1]). Suppose that a theory has the following properties: (1) it is monoentropic, meaning that measurement entropy equals mixing entropy for all systems, (2) its measurement entropy is strongly subadditive and (3) its measurement entropy satisfies the Holevo bound. Then the theory satisfies information causality. It follows that any theory satisfying these conditions cannot violate Tsirel’son’s bound. In [1] we noted that monoentropicity was only used to establish the technical condition of positivity of conditional information, H(A|C), defined as H(AC)− H(C), when A is classical. This condition does not hold for all theories in the broad class we consider. Shortly after submitting the final version of [1], but too late for inclusion therein, we noticed that it does, however, hold for all theories satisfying strong subadditivity, enabling us to strengthen our theorem as follows. Theorem 2. Suppose that a theory’s measurement entropy is strongly subadditive and satisfies the Holevo bound. Then the theory satisfies information causality. It follows that any theory satisfying these conditions cannot violate Tsirel’son’s bound. As noted in our earlier paper, one can replace the two premises of this theorem with the single premise that the measurement entropy satisfies the data processing inequality. To see that H(A|C)> 0 for classical A follows from strong subadditivity, we recall (from e.g. lemma 1 of [1]), that strong subadditivity is equivalent to the positivity of conditional mutual information, I (A : B|C)> 0. Letting A and B be perfectly correlated classical systems in this inequality, we have I (A : B|C)= H(A|C)+ H(B|C)− H(AB|C) = H(AC)− H(C)+ H(BC) − H(C)− H(ABC)+ H(C) = H(AC)+ H(BC)− H(ABC)− H(C). New Journal of Physics 14 (2012) 129401 (http://www.njp.org/)

•Jan 1, 2012•0 citations

Notes on the equality in SSA of entropy on CAR algebra

József Pitrik, V. P. Belavkin

We prove a necessary an sufficient condition for the states which satisfy strong subadditivity of von Neumann entropy with equality on CAR algebra and we show an example when the equality holds but the state is not separable.

arXiv:math-ph/0602035•Feb 13, 2006•1 citation

Towards a Topological Proof of the Strong Subadditivity

Chih-Yu Lo, Po-Yao Chang

Topological entanglement entropy (TEE) represents an intrinsic contribution to the entanglement entropy (EE) in topologically ordered systems. In quantum information theory, strong subadditivity (SSA) is a fundamental property of EE, reflecting the non-negativity of conditional mutual information. TEE was originally believed to be a universal correction to the area law of EE, suggesting that its SSA would directly follow from the SSA of EE. However, due to spurious contributions, the correction term is not universal; consequently, the value predicted by topological quantum field theory (TQFT) provides only a lower bound. In this work, we present a topological analysis showing that the SSA for TEE is equivalent to a specific inequality within the TQFT framework. We further verify that this inequality holds for all known unitary modular tensor categories (UMTCs) up to rank 11, supporting the conjecture that SSA holds universally in the TQFT framework. Conversely, assuming the validity of the SSA condition, the inequality can be interpreted as a consistency criterion for candidate UMTCs.

arXiv:2411.05077•Nov 7, 2024•0 citations

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
We prove that strong subadditibity(SSA) of entropy holds for fermion systems. Main resultProbably,strong subadditibity(SSA) is the one of the most powerful prop- erties of entropy.In this paper, we prove SSA holds for fermion systems.We consider Fermonic systems on an integer lattice Zd,the\mathrm{Z}^{d},\mathrm{t}\mathrm{h}\mathrm{e}Zd,the set of d\mathrm{d}d -tuples of integers.The creation and annihilation operators will be denoted by aj∗a_{j}^{*}aj∗​ and aj(j∈Zd)a_{j}(j\in \mathrm{Z}^{d})aj​(j∈Zd) .They satisfy the canonical anticommutation relations,
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.
In this addendum to our paper (2010 New J. Phys. 12 033024), we point out that an elementary consequence of the strong subadditivity inequality allows us to strengthen one of the main conclusions of that paper. In this addendum to our paper [1], we point out that an elementary consequence of the strong subadditivity inequality allows us to strengthen one of the main conclusions of that paper. In [1], we defined notions of measurement and mixing entropy for a broad class of probabilistic theories (which includes finite-dimensional quantum theory). We defined a theory to be monoentropic if for every state of a system in the theory, the mixing entropy (denoted S) is equal to the measurement entropy (denoted H ). This broad class of theories has a notion of composite system, and of the ‘marginal state’ of a subsystem given a state for the composite system, so that the strong subadditivity inequality may be formulated for real-valued functions of states, including the measurement and mixing entropies. We proved the following theorem. Theorem 1 (theorem 4 of [1]). Suppose that a theory has the following properties: (1) it is monoentropic, meaning that measurement entropy equals mixing entropy for all systems, (2) its measurement entropy is strongly subadditive and (3) its measurement entropy satisfies the Holevo bound. Then the theory satisfies information causality. It follows that any theory satisfying these conditions cannot violate Tsirel’son’s bound. In [1] we noted that monoentropicity was only used to establish the technical condition of positivity of conditional information, H(A|C), defined as H(AC)− H(C), when A is classical. This condition does not hold for all theories in the broad class we consider. Shortly after submitting the final version of [1], but too late for inclusion therein, we noticed that it does, however, hold for all theories satisfying strong subadditivity, enabling us to strengthen our theorem as follows. Theorem 2. Suppose that a theory’s measurement entropy is strongly subadditive and satisfies the Holevo bound. Then the theory satisfies information causality. It follows that any theory satisfying these conditions cannot violate Tsirel’son’s bound. As noted in our earlier paper, one can replace the two premises of this theorem with the single premise that the measurement entropy satisfies the data processing inequality. To see that H(A|C)> 0 for classical A follows from strong subadditivity, we recall (from e.g. lemma 1 of [1]), that strong subadditivity is equivalent to the positivity of conditional mutual information, I (A : B|C)> 0. Letting A and B be perfectly correlated classical systems in this inequality, we have I (A : B|C)= H(A|C)+ H(B|C)− H(AB|C) = H(AC)− H(C)+ H(BC) − H(C)− H(ABC)+ H(C) = H(AC)+ H(BC)− H(ABC)− H(C). New Journal of Physics 14 (2012) 129401 (http://www.njp.org/)
Topological entanglement entropy (TEE) represents an intrinsic contribution to the entanglement entropy (EE) in topologically ordered systems. In quantum information theory, strong subadditivity (SSA) is a fundamental property of EE, reflecting the non-negativity of conditional mutual information. TEE was originally believed to be a universal correction to the area law of EE, suggesting that its SSA would directly follow from the SSA of EE. However, due to spurious contributions, the correction term is not universal; consequently, the value predicted by topological quantum field theory (TQFT) provides only a lower bound. In this work, we present a topological analysis showing that the SSA for TEE is equivalent to a specific inequality within the TQFT framework. We further verify that this inequality holds for all known unitary modular tensor categories (UMTCs) up to rank 11, supporting the conjecture that SSA holds universally in the TQFT framework. Conversely, assuming the validity of the SSA condition, the inequality can be interpreted as a consistency criterion for candidate UMTCs.
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.