Unambiguous comparison of quantum measurements

TL;DR

This paper investigates methods for unambiguously comparing unknown quantum measurements, distinguishing between labeled and unlabeled outcomes, and provides optimal strategies and success probabilities for different scenarios.

Contribution

It formulates the unambiguous comparison problem for quantum measurements, analyzes optimal strategies, and derives success probabilities for both labeled and unlabeled cases.

Findings

Success probability decreases as 1/d with Hilbert space dimension for labeled outcomes.

Single-shot comparison is impossible for unlabeled outcomes in 2D; at least two uses are needed.

Optimal test state achieves 75% success probability in 2D unlabeled case.

Abstract

The goal of comparison is to reveal the difference of compared objects as fast and reliably as possible. In this paper we formulate and investigate the unambiguous comparison of unknown quantum measurements represented by non-degenerate sharp POVMs. We distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases we can unambiguously conclude only that the measurements are different. For the labeled case it is sufficient to use each unknown measurement only once and the average conditional success probability decreases with the Hilbert space dimension as 1/d. If the outcomes of the apparatuses are not labeled, then the problem is more complicated. We analyze the case of two-dimensional Hilbert space. In this case single shot comparison is impossible and each measurement device must be used (at least) twice. The optimal test state in the two-shots scenario gives the average conditional success probability 3/4. Interestingly, the optimal experiment detects unambiguously the difference with nonvanishing probability for any pair of observables.