Near-optimal performance of square-root measurement for general score functions and quantum ensembles

TL;DR

This paper generalizes the Barnum-Knill theorem and the pretty good measurement concept to continuous and infinite-dimensional quantum ensembles, showing near-optimal performance in Bayesian estimation tasks.

Contribution

It extends the pretty good measurement framework and Barnum-Knill theorem to continuous and infinite-dimensional quantum ensembles, introducing a new performance metric.

Findings

Generalized pretty good measurement performs within a factor of two of the optimal in Bayesian mean square error.

Introduced a new performance metric based on expected gain for quantum measurements.

Extended the theoretical understanding of quantum state discrimination beyond finite ensembles.

Abstract

The Barnum-Knill theorem states that the optimal success probability in the multiple state discrimination task is not more than the square root of the success probability when the pretty good or square-root measurement is used for this task. An assumption of the theorem is that the underlying ensemble consists of finitely many quantum states over a finite-dimensional quantum system. Motivated in part by the fact that the success probability is not a relevant metric for continuous ensembles, in this paper we provide a generalization of the notion of pretty good measurement and the Barnum-Knill theorem for general quantum ensembles, including those described by a continuous parameter space and an infinite-dimensional Hilbert space. To achieve this, we also design a general metric of performance for quantum measurements that generalizes the success probability, namely, the expected gain of the measurement with respect to a positive score function. A notable consequence of the main result is that, in a Bayesian estimation task, the mean square error of the generalized pretty good measurement does not exceed twice the optimal mean square error.