Error rates of Belavkin weighted quantum measurements and a converse to Holevo's asymptotic optimality theorem

TL;DR

This paper analyzes the performance of Belavkin weighted quantum measurements, demonstrating the superiority of quadratic weighting over PGM in two-state discrimination and establishing conditions for asymptotic optimality.

Contribution

It proves that quadratic weighting unconditionally outperforms PGM for two states and provides a converse to Holevo's asymptotic optimality theorem, with counterexamples for three states.

Findings

Quadratic weighting outperforms PGM in two-state discrimination.

A converse theorem shows weighted measurements are asymptotically optimal only if quadratically weighted.

Counterexamples demonstrate limitations for three-state ensembles.

Abstract

We compare several instances of pure-state Belavkin weighted square-root measurements from the standpoint of minimum-error discrimination of quantum states. The quadratically weighted measurement is proven superior to the so-called "pretty good measurement" (PGM) in a number of respects: (1) Holevo's quadratic weighting unconditionally outperforms the PGM in the case of two-state ensembles, with equality only in trivial cases. (2) A converse of a theorem of Holevo is proven, showing that a weighted measurement is asymptotically optimal only if it is quadratically weighted. Counterexamples for three states are constructed. The cube-weighted measurement of Ballester, Wehner, and Winter is also considered. Sufficient optimality conditions for various weights are compared.