On Quantum Detection and the Square-Root Measurement

TL;DR

This paper introduces a new characterization of the square-root measurement (SRM) for quantum state discrimination, proving its optimality in least-squares sense and error minimization for geometrically uniform states.

Contribution

It provides a novel characterization of the SRM and demonstrates its optimality in least-squares and error probability for specific quantum state sets.

Findings

SRM is optimal in a least-squares sense.

SRM minimizes detection error for geometrically uniform states.

New theoretical insights into quantum measurement optimization.

Abstract

In this paper we consider the problem of constructing measurements optimized to distinguish between a collection of possibly non-orthogonal quantum states. We consider a collection of pure states and seek a positive operator-valued measure (POVM) consisting of rank-one operators with measurement vectors closest in squared norm to the given states. We compare our results to previous measurements suggested by Peres and Wootters [Phys. Rev. Lett. 66, 1119 (1991)] and Hausladen et al. [Phys. Rev. A 54, 1869 (1996)], where we refer to the latter as the square-root measurement (SRM). We obtain a new characterization of the SRM, and prove that it is optimal in a least-squares sense. In addition, we show that for a geometrically uniform state set the SRM minimizes the probability of a detection error. This generalizes a similar result of Ban et al. [Int. J. Theor. Phys. 36, 1269 (1997)].