Completeness Stability of Quantum Measurements

TL;DR

This paper introduces a new resource monotone called completeness stability to evaluate and optimize quantum measurements, linking stability to well-known measurement designs like SIC-POVMs.

Contribution

It defines a novel stability measure for quantum measurements, connects it to frame theory, and characterizes optimal measurements as weighted complex projective 2-designs.

Findings

The minimum eigenvalue of a frame operator quantifies measurement stability.

Maximizing this monotone identifies measurements as weighted complex projective 2-designs.

Includes well-known measurement types such as SIC-POVMs.

Abstract

We introduce a resource monotone, the completeness stability, to quantify the quality of quantum measurements within a resource-theoretic framework. By viewing a quantum measurement as a frame, the minimum eigenvalue of a frame operator emerges as a significant monotone. It captures bounds on estimation errors and the numerical stability of inverting the frame operator to calculate the optimal dual for state reconstruction. Maximizing this monotone identifies a well-characterized class of quantum measurements forming weighted complex projective 2-designs, which includes well-known examples such as SIC-POVMs. Our results provide a principled framework for comparing and optimizing quantum measurements for practical applications.