On the Optimality of Square Root Measurements in Quantum State Discrimination

TL;DR

This paper investigates when square root measurements are optimal for quantum state discrimination, providing necessary and sufficient conditions and applying findings to quantum communication schemes.

Contribution

It establishes precise conditions under which square root measurements maximize correct discrimination probability for GUS state sets.

Findings

Derived necessary and sufficient conditions for SRM optimality.

Applied conditions to quantum PSK and PPM state constellations.

Enhanced understanding of measurement strategies in quantum communication.

Abstract

Distinguishing assigned quantum states with assigned probabilities via quantum measurements is a crucial problem for the transmission of classical information through quantum channels. Measurement operators maximizing the probability of correct discrimination have been characterized by Helstrom, Holevo and Yuen since 1970's. On the other hand, closed--form solutions are available only for particular situations enjoying high degrees of symmetry. As a suboptimal solution to the problem, measurement operators, directly determined from states and probabilities and known as square root measurements (SRM), were introduced by Hausladen and Wootters. These operators were also recognized to be optimal for pure states equipped with geometrical uniform symmetry (GUS). In this paper we discuss the optimality of the SRM and find necessary and sufficient conditions in order that SRM maximize the correct decision probabilities for set of states formed by several constellations of GUS states. The results are applied to some specific examples concerning double constellations of quantum phase shift keying (PSK) and pulse position modulation (PPM) states, with possible applications to practical systems of quantum communications.