Doubly minimized Petz and sandwiched Renyi mutual information: Operational interpretation from binary quantum state discrimination

TL;DR

This paper provides an operational interpretation of doubly minimized Petz and sandwiched Renyi mutual information in quantum state discrimination, linking these measures to error exponents and asymptotics.

Contribution

It establishes the connection between these quantum mutual information measures and the operational tasks of correlation detection and error exponent analysis.

Findings

Direct exponent determined by Petz Renyi mutual information for $eta o 1/2$

Strong converse exponent determined by sandwiched Renyi mutual information for $eta o 1$

Includes analysis of moderate deviations, Stein exponent, and second-order asymptotics.

Abstract

The doubly minimized Petz Renyi mutual information of order $\alpha$ is defined as the minimum of the Petz divergence of order $\alpha$ of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Renyi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Renyi mutual information of order $\alpha\in (1/2,1)$, and that the strong converse exponent is determined by the doubly minimized sandwiched Renyi mutual information of order $\alpha\in (1,\infty)$. This provides an operational interpretation of these types of Renyi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.