Pretty good measures in quantum information theory

Abstract

Quantum generalizations of Rényi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Rényi divergence D̅ α and the minimal quantum Rényi divergence D̅ α . In this paper, we prove a reverse Araki-Lieb-Thirring inequality that implies a new relation between these two families of divergences, namely that αD̅ α (ρ∥σ) ≤ D̅ α (ρ∥σ) for α ϵ [0, 1] and where ρ and σ are density operators. This bound suggests defining a “pretty good fidelity”, whose relation to the usual fidelity implies the known relations between the optimal and pretty good measurement as well as the optimal and pretty good singlet fraction.