The proper formula for relative entropy and its asymptotics in quantum probability

Abstract

Umegaki's relative entropy S(ω,φ) = TrD ω (\ogD ω -\ogD φ ) (of states ω and φ with density operators D ω and D φ , respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality ΎT A log AB is obtained for positive definite matrices A and B.

An Ergodic Theorem for the Quantum Relative Entropy

“…This was conjectured already in Hiai/Petz [3] and proved for the case that both states are product (i.i.d.) states by Ogawa and Nagaoka in [4].…”

“…An important ingredient in our proof of the Theorem 2.1 is a result proved by Hiai and Petz in [3]. The starting point is the spectral decomposition of the density operator D ϕ (1) corresponding to the state ϕ (1) on A:…”

“…The methods used to derive the results are mainly based on the techniques developed in the paper Hiai/Petz [3] and in [1]. The Hiai/Petz paper considered the class of completely ergodic states ψ, and only an (exact) upper bound for the separation order was derived.…”

Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

“…Indeed, by the contractivity of the trace distance under quantum operations (compare Thm. 9.2 in [29]) and by assumption (17), it holds…”

“…The quantum extension of the Shannon-McMillan Theorem was first obtained in [19] for Bernoulli sources, then a partial assertion was obtained for the restricted class of completely ergodic sources in [17], and finally in [7], a complete quantum extension was shown for general ergodic sources. The latter result is based on the construction of subspaces of dimension close to 2 ns , being typical for the source, in the sense that for sufficiently large block length n, their corresponding orthogonal projectors have an expectation value arbitrarily close to 1 with respect to the state of the quantum source.…”