Optimal Quantum Measurements with respect to the Fidelity

TL;DR

This paper investigates the structure of fidelity-optimal quantum measurements, revealing conditions for uniqueness or multiplicity, and provides a complete characterization in the case of pure states using geometric methods.

Contribution

It clarifies the structure of minimal fidelity-optimal measurements, especially for singular states, and characterizes all such measurements when one state is pure.

Findings

Existence of either a unique or infinitely many minimal optimal measurements.

Uniqueness holds under a weak commutativity condition.

Complete characterization of minimal optimal measurements for pure states.

Abstract

Fidelity is the standard measure for quantifying the similarity between two quantum states. It is equal to the square of the minimum Bhattacharyya coefficient between the probability distributions induced by quantum measurements on the two states. Though established for over thirty years, the structure of fidelity-optimal quantum measurements remains unclear when the two density operators are singular (not invertible). Here we address this gap, with a focus on minimal optimal measurements, which admit no nontrivial coarse graining that is still optimal. We show that there exists either a unique minimal optimal measurement or infinitely many inequivalent choices. Moreover, the first case holds if and only if the two density operators satisfy a weak commutativity condition. In addition, we provide a complete characterization of all minimal optimal measurements when one state is pure, leveraging geometric insights from the Bloch-sphere representation. The connections with quantum incompatibility, operator pencils, and geometric means are highlighted.