Quantum Chi-Squared and Goodness of Fit Testing

TL;DR

This paper explores quantum hypothesis testing, focusing on the statistical fluctuations in experiments, and introduces a quantum chi-squared divergence for assessing quantum state fidelity with optimized measurement strategies.

Contribution

It introduces a quantum chi-squared divergence, analyzes optimal measurement bases, and characterizes the efficiency of quantum goodness of fit tests.

Findings

Quantum chi-squared divergence derived from classical chi-squared test.

Optimal measurement basis maximizes Pitman and Bahadur efficiency.

Exponential scaling of confidence in quantum hypothesis testing.

Abstract

The density matrix in quantum mechanics parameterizes the statistical properties of the system under observation, just like a classical probability distribution does for classical systems. The expectation value of observables cannot be measured directly, it can only be approximated by applying classical statistical methods to the frequencies by which certain measurement outcomes (clicks) are obtained. In this paper, we make a detailed study of the statistical fluctuations obtained during an experiment in which a hypothesis is tested, i.e. the hypothesis that a certain setup produces a given quantum state. Although the classical and quantum problem are very much related to each other, the quantum problem is much richer due to the additional optimization over the measurement basis. Just as in the case of classical hypothesis testing, the confidence in quantum hypothesis testing scales exponentially in the number of copies. In this paper, we will argue 1) that the physically relevant data of quantum experiments is only contained in the frequencies of the measurement outcomes, and that the statistical fluctuations of the experiment are essential, so that the correct formulation of the conclusions of a quantum experiment should be given in terms of hypothesis tests, 2) that the (classical) $\chi^2$ test for distinguishing two quantum states gives rise to the quantum $\chi^2$ divergence when optimized over the measurement basis, 3) present a max-min characterization for the optimal measurement basis for quantum goodness of fit testing, find the quantum measurement which leads both to the maximal Pitman and Bahadur efficiency, and determine the associated divergence rates.