Mary Beth Ruskai
Published 2002-05-13, updated 2002-05-19Version 2
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.
Comments: 28 pages, latex Added reference to M.J.W. Hall, "Quantum Information and Correlation Bounds" Phys. Rev. A, 55, pp 100--112 (1997)
Journal: J. Math. Phys. 43, 4358-4375 (2002); erratum 46, 019901 (2005).
Bounds on the Concavity of Quantum Entropy
Characterizing the dynamical semigroups that do not decrease a quantum entropy
On the conditions for the classicality of a quantum particle
