Inequalities for quantum entropy: A review with conditions for equality

By Mary Beth Ruskai. 2002

237 citations recorded by the source corpus.

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 the...

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/54e1be53b4e5fed17ab374ebfbc2ba1ddc794d57

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistical and Nonlinear Physics

Citations in the corpus

Fundamental properties of Tsallis relative entropy (2004, background): “Remark 2.7 It is known [20] (see also [ 25 ]) that there is an equivalent relation between the monotonicity for the quantum relative entropy and the strong subadditivity for the quantum entropy.”
Monotonicity of quantum relative entropy revisited (2002, methodology): “Condition (2) of Theoem 2 appears also in the paper [22] in which different methods are used.”
On quantum Rényi entropies: A new generalization and some properties (2013, background): “The quantum relative entropy of ρ and σ is deﬁned as We refer to [33] for a review of properties of this quantity.”
Geometry of Quantum States: Index (2007, background): “7 For a proof – in fact several proofs – and more information on entropy inequalities generally, we recommend two reviews written by experts, Lieb (1975) [600] and Ruskai (2002) [784].”
Monotonicity of the Quantum Relative Entropy Under Positive Maps (2015, background): “The question of monotonicity of the quantum relative entropy has however remained open [12, 30, 34, 36] for the natural class of linear maps Φ : T (H) → T (H′) that are subject only to the constraint that they map quantum states to quantum states.”
User-Friendly Tail Bounds for Sums of Random Matrices (2010, background): “II], and Ruskai offers a simplified account of Epstein’s argument in [47].”