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arXiv:0802.1234 · Proceedings of the National Academy of Sciences

A matrix convexity approach to some celebrated quantum inequalities

Edward G. Effros

Published Jan 21, 2009• 145 citations
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Abstract

Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequalities. A general argument is given using matrix perspectives of operator convex functions. A matrix analogue of Maréchal's extended perspectives provides additional inequalities, including a p + q < or = 1 result of Lieb.

DOI
10.1073/pnas.0807965106

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Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequalities. A general argument is given using matrix perspectives of operator convex functions. A matrix analogue of Maréchal's extended perspectives provides additional inequalities, including a p + q < or = 1 result of Lieb.
Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequality. A general argument is given in terms of the matrix perspective of an operator convex function. A matrix analogue of Maréchal's extended perspectives provides additional inequalities, including a p+q≤1p+q\leq 1p+q≤1 result of Lieb.
Just as the classical f-divergences parameterized on convex functions generalize the classical relative entropy, the quantum quasi-entropies generalize the quantum relative entropy parameterized on operator convex functions. Quantum quasi-entropies satisfy a version of the monotonicity property and are jointly convex in their arguments. We provide the equality conditions for these inequalities for a family of operator convex functions that includes the von Neumann entropies as a special case. Quantum quasi-entropies are defined with an arbitrarily chosen matrix and since the inequalities are true for any choice of such matrices, we show that these inequalities can be interpreted as operator inequalities.
This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy 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 one of Lieb's less well-known concave trace functions, allows one to obtain conditions for equality. 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 paper concludes with an Appendix giving a short description of Epstein's elegant proof of the relevant concavity theorem of Lieb.
We consider both known and not previously studied trace functions with applications in quantum physics. By using perspectives we obtain convexity statements for different notions of residual entropy, including the entropy gain of a quantum channel as studied by Holevo and others. We give new and simplified proofs of the Carlen-Lieb theorems concerning concavity or convexity of certain trace functions by making use of the theory of operator monotone functions. We then apply these methods in a study of new types of trace functions. Keywords: Trace function, convexity, entropy gain, residual entropy, operator monotone function.