Worst-case Quantum Hypothesis Testing with Separable Measurements

TL;DR

This paper investigates the limits of quantum hypothesis testing using separable measurements, providing analytical and numerical solutions, especially for two-qubit systems, and highlighting the impact on quantum state verification.

Contribution

It formulates the worst-case hypothesis testing problem as an SDP, analyzes the two-qubit case in detail, and explores the limitations of separable measurements in quantum state discrimination.

Findings

SDP formulation for worst-case hypothesis testing

Analytical solutions for commuting and orthogonal hypotheses

Separable measurements often prevent perfect distinguishability

Abstract

For any pair of quantum states (the hypotheses), the task of binary quantum hypotheses testing is to derive the tradeoff relation between the probability $p_{01}$ of rejecting the null hypothesis and $p_{10}$ of accepting the alternative hypothesis. The case when both hypotheses are explicitly given was solved in the pioneering work by Helstrom. Here, instead, for any given null hypothesis as a pure state, we consider the worst-case alternative hypothesis that maximizes $p_{10}$ under a constraint on the distinguishability of such hypotheses. Additionally, we restrict the optimization to separable measurements, in order to describe tests that are performed locally. The case $p_{01}=0$ has been recently studied under the name of "quantum state verification". We show that the problem can be cast as a semi-definite program (SDP). Then we study in detail the two-qubit case. A comprehensive study in parameter space is done by solving the SDP numerically. We also obtain analytical solutions in the case of commuting hypotheses, and in the case where the two hypotheses can be orthogonal (in the latter case, we prove that the restriction to separable measurements generically prevents perfect distinguishability). In regards to quantum state verification, our work shows the existence of more efficient strategies for noisy measurement scenarios.