Error-Minimizing Measurements in Postselected One-Shot Symmetric Quantum State Discrimination and Acceptance as a Performance Metric

TL;DR

This paper explores optimal measurement strategies in postselected quantum hypothesis testing, characterizing all minimum-error measurements, introducing an acceptance metric, and identifying the best measurements in terms of acceptance.

Contribution

It provides a comprehensive characterization of all minimum-error measurements in postselected quantum hypothesis testing and introduces an acceptance metric to evaluate measurement quality.

Findings

Derived the set of all minimum-error measurements.

Introduced an acceptance metric for postselected testing.

Identified measurements that maximize acceptance among minimum-error strategies.

Abstract

In hypothesis testing with quantum states, given a black box containing one of the two possible states, measurement is performed to detect in favor of one of the hypotheses. In postselected hypothesis testing, a third outcome is added, corresponding to not selecting any of the hypotheses. In postselected scenario, minimum error one-shot symmetric hypothesis testing is characterized in literature conditioned on the fact that one of the selected outcomes occur. We proceed further in this direction to give the set of all possible measurements that lead to the minimum error. We have given an arbitrary error-minimizing measurement in a parametric form. Note that not selecting any of the hypotheses decimates the quality of testing. We further give an example to show that these measurements vary in quality. There is a need to discuss the quality of postselected hypothesis testing. We then characterize the quality of postselected hypothesis testing by defining a new metric acceptance and give expression of acceptance for an arbitrary error-minimizing measurement in terms of some parameters of the measurement. On the set of measurements that achieve minimum error, we have maximized the acceptance, and given an example which achieves that, thus giving an example of the best possible measurement in terms of acceptance.