Some continuity properties of quantum Rényi divergences

Abstract

In the problem of binary quantum channel discrimination with product inputs, the supremum of all type II error exponents for which the optimal type I errors go to zero is equal to the Umegaki channel relative entropy, while the infimum of all type II error exponents for which the optimal type I errors go to one is equal to the infimum of the sandwiched channel Rényi $\alpha$-divergences over all $\alpha >1$. We prove the equality of these two threshold values (and therefore the strong converse property for this problem) using a minimax argument based on a newly established continuity property of the sandwiched Rényi divergences. Motivated by this, we give a detailed analysis of the continuity properties of various other quantum (channel) Rényi divergences, which may be of independent interest.

v4: Continuity is studied on more general sets of the form $\rho \le f(\sigma )$ for a large class of functions $f$. 44 pages

References in corpus (4)

• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• The Converse Part of The Theorem for Quantum Hoeffding Bound
• Test-measured Rényi divergences

Cited by in corpus (8)

• Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond
• Geometric relative entropies and barycentric Rényi divergences
• Locally-Measured Rényi Divergences
• Sufficiency of Rényi divergences
• Uhlmann's theorem for relative entropies
• Uhlmann's theorem for measured divergences
• One-shot Multiple Access Channel Simulation
• Log-majorizations between quasi-geometric type means for matrices