Quantum hypothesis testing for quantum Gaussian states: Quantum analogues of chi-square, t and F tests

TL;DR

This paper develops quantum analogues of classical statistical tests like chi-square, t, and F tests for quantum Gaussian states, providing optimal testing procedures in quantum hypothesis testing.

Contribution

It introduces a general reduction theorem that simplifies complex quantum hypothesis testing problems into fundamental ones, enabling the derivation of quantum chi-square, t, and F tests.

Findings

Quantum counterparts of chi-square, t, and F tests are derived as optimal tests.

A general reduction theorem simplifies complex quantum hypothesis testing problems.

The methods incorporate group symmetry and the quantum Hunt-Stein theorem.

Abstract

We treat quantum counterparts of testing problems whose optimal tests are given by chi-square, t and F tests. These quantum counterparts are formulated as quantum hypothesis testing problems concerning quantum Gaussian states families, and contain disturbance parameters, which have group symmetry. Quantum Hunt-Stein Theorem removes a part of these disturbance parameters, but other types of difficulty still remain. In order to remove them, combining quantum Hunt-Stein theorem and other reduction methods, we establish a general reduction theorem that reduces a complicated quantum hypothesis testing problem to a fundamental quantum hypothesis testing problem. Using these methods, we derive quantum counterparts of chi-square, t and F tests as optimal tests in the respective settings.