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Reviews in Mathematical Physics

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

Mary Beth Ruskai

Published Jan 1, 1994• 161 citations
Publisher

Abstract

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.

DOI
10.1142/s0129055x94000407

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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
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.
We propose a conceptually simple method for proving various entropic inequalities by means of spectral estimates. Our approach relies on a two-sided estimate of the relative entropy between two quantum states in terms of certain variance-type quantities. As an application of this estimate, we answer three important problems in quantum information theory and dissipative quantum systems that were left open until now: first, we prove the existence of strictly contractive constants for the strong data processing inequality for quantum channels that satisfy a notion of detailed balance. Second, we prove the existence of the complete modified logarithmic Sobolev constant which controls the exponential entropic convergence of a quantum Markov semigroup towards its equilibrium. Third, we prove a new generalization of the strong subadditivity of the entropy for the relative entropy distance to a von Neumann algebra under a specific gap condition. All the three results obtained are independent of the size of the environment.
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.
We consider a quantum quasi-relative entropy [Formula: see text] for an operator [Formula: see text] and an operator convex function [Formula: see text]. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the [Formula: see text]-divergences (i.e. [Formula: see text]). We also provide an error term for a class of operator inequalities, that generalizes operator strong subadditivity inequality. We apply those results to demonstrate explicit bounds for the logarithmic function, that leads to the quantum relative entropy, and the power function, which gives, in particular, a Wigner–Yanase–Dyson skew information. In particular, we provide the remainder terms for the strong subadditivity inequality, operator strong subadditivity inequality, WYD-type inequalities, and the Cauchy–Schwartz inequality.
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/)