The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states

TL;DR

This paper explores the quantum Chernoff bound as a measure of how distinguishable two quantum states are, introducing new metrics and applying them to qubit and Gaussian states.

Contribution

It defines a new quantum distinguishability measure based on the Chernoff bound and relates it to the Wigner-Yanase metric, with practical implementations for qubits and Gaussian states.

Findings

The quantum Chernoff bound provides a meaningful measure of state distinguishability.

The new metric coincides with the Wigner-Yanase metric.

Application to qubits and Gaussian states demonstrates practical relevance.

Abstract

Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. [Phys. Rev. Lett. 98, 160501] and Nussbaum and Szkola [quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on the quantum Chernoff bound, we define a physically meaningful distinguishability measure and its corresponding metric in the space of states; the latter is shown to coincide with the Wigner-Yanase metric. Along the same lines, we define a second, more easily implementable, distinguishability measure based on the error probability of discrimination when the same local measurement is performed on every copy. We study some general properties of these measures, including the probability distribution of density matrices, defined via the volume element induced by the metric, and illustrate their use in the paradigmatic cases of qubits and Gaussian infinite-dimensional states.