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

TL;DR

This work generalizes the pretty good measurement to arbitrary quantum ensembles, including continuous parameter spaces and infinite-dimensional systems, by introducing the generalized pretty good measurement (GPGM) and a score-based performance metric—the expected gain with respect to positive score functions. The authors prove a generalized Barnum–Knill bound showing the optimal expected gain is at most the square root of the gain achieved by the GPGM, and they establish a near-optimality result for Bayesian mean squared error, with the GPGM MSE bounded by a factor of two from the optimum. They develop two key mathematical results for integrating real-valued functions with operator-valued measures, enabling rigorous proofs of the main theorems. The framework yields practical, theoretically grounded guarantees for quantum parameter estimation, including bosonic Gaussian ensembles, and provides a unified approach to discrimination, estimation, and measurement design in broad quantum settings.

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.

Theorems & Definitions (14)

• Definition 3.1: Generalized pretty good measurement
• Theorem 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.2
• Corollary 3.3: Barnum--Knill theorem
• Lemma 3.4
• Proposition 3.5
• Theorem 3.6
• Theorem 1.1