Multiple Quantum Hypothesis Testing Expressions and Classical-Quantum Channel Converse Bounds

TL;DR

This paper derives new exact expressions for quantum hypothesis testing error probabilities and demonstrates their implications for classical-quantum channel bounds, confirming the tightness of existing converse bounds.

Contribution

It introduces alternative exact formulas for quantum hypothesis testing error probabilities and applies them to establish the tightness of classical-quantum channel converse bounds.

Findings

Derived two new exact expressions for quantum hypothesis testing error probabilities.

Proved the tightness of Matthews-Wehner and Hayashi-Nagaoka converse bounds.

Unified classical-quantum channel analysis using these new expressions.

Abstract

Alternative exact expressions are derived for the minimum error probability of a hypothesis test discriminating among $M$ quantum states. The first expression corresponds to the error probability of a binary hypothesis test with certain parameters; the second involves the optimization of a given information-spectrum measure. Particularized in the classical-quantum channel coding setting, this characterization implies the tightness of two existing converse bounds; one derived by Matthews and Wehner using hypothesis-testing, and one obtained by Hayashi and Nagaoka via an information-spectrum approach.