Operating Characteristics for Binary Hypothesis Testing in Quantum Systems

TL;DR

This paper introduces quantum analogs of classical ROC curves for binary hypothesis testing, providing a unified framework and new measurement strategies to optimize quantum discrimination tasks.

Contribution

It proposes decision and measurement operating characteristics for quantum hypothesis testing, generalizes classical-quantum measurement correspondence, and offers a constructive method for designing measurements beyond Helstrom's optimal.

Findings

Introduces quantum ROC-like curves for hypothesis testing

Generalizes classical-quantum measurement relationships

Provides a method to construct various minimum-error measurements

Abstract

Receiver operating characteristics (ROCs) are a well-established representation of the tradeoff between detection and false alarm probabilities in classical binary hypothesis testing. We use classical ROCs as motivation for two types of operating characteristics for binary hypothesis testing in quantum systems -- decision operating characteristics (QDOCs) and measurement operating characteristics (QMOCs). Both are described in the context of a framework we propose that encompasses the typical formulations of binary hypothesis testing in both the classical and quantum scenarios. We interpret Helstrom's well-known result regarding discrimination between two quantum density operators with minimum probability of error in this framework. We also present a generalization of previous results regarding the correspondence between classical Parseval frames and quantum measurements. The derivation naturally leads to a constructive procedure for generating many different measurements besides Helstrom's optimal measurement, some standard and others non-standard, that achieve minimum probability of error.