Abstract
We present an in-depth study of the problem of multiple-shot discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: minimum error and unambiguous discrimination. In the case of minimum error discrimination, we focus on discrimination of measurements with the assistance of entanglement. We provide an alternative proof of the fact that all pairs of distinct von Neumann measurements can be distinguished perfectly (i.e. with the unit success probability) using only a finite number of queries. Moreover, we analytically find the minimal number of queries needed for perfect discrimination. We also show that in this scenario querying the measurements in parallel gives the optimal strategy, and hence any possible adaptive methods do not offer any advantage over the parallel scheme. In the unambiguous discrimination scenario, we give the general expressions for the optimal discrimination probabilities with and without the assistance of entanglement. Finally, we show that typical pairs of Haar-random von Neumann measurements can be perfectly distinguished with only two queries.
Highlights
With the recent technological progress, quantum information science is not merely a collection of purely theoretical ideas anymore
We have presented a comprehensive treatment of the problem of discrimination of von Neumann measurements
We showed an alternative proof of the fact that for any pair of measurements P1 and P2, P1 = P2, there exists a finite number N of uses of the black box which allows us to achieve perfect discrimination
Summary
With the recent technological progress, quantum information science is not merely a collection of purely theoretical ideas anymore. In a recent demonstration of quantum computational supremacy (advantage) by the collaboration of Google and UCBS [14] the researchers reported single-qubit measurement errors that are of order of a few percents This motivates the interest in certification strategies tailored to von-Neumann measurements. The results contained in this work concern the following two scenarios: Minimum error discrimination— In this setting, we are allowed to use the black box containing von Neumann measurement many times. We can prepare any input state with an arbitrarily large quantum memory (i.e., we can use ancillas of arbitrarily large dimension), and we can perform any channels between usages of the black box This allows us to implement both parallel (see Fig. 1) as well as adaptive discrimination strategies (see Fig. 2).
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