Multiple-shot and unambiguous discrimination of von Neumann measurements

Zbigniew Puchała1,2, Łukasz Pawela1, Aleksandra Krawiec1,3, Ryszard Kukulski1,4, and Michał Oszmaniec5,6

1Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul. Bałtycka 5, 44-100 Gliwice, Poland
2Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
3Institute of Mathematics, Silesian University of Technology, ul. Kaszubska 23, 44-100 Gliwice, Poland
4Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland
5Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, ul. Wita Stwosza 57, 80-308 Gdańsk, Poland
6Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland

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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 $\textit{in parallel}$ 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.

We are interested in the following problem. Imagine we have an unknown device hidden in a black box. We know it performs one of the two possible von Neumann measurements. Generally, whenever a quantum state is sent through the box, the box produces, with probabilities predicted by quantum mechanics, classical labels corresponding to the measurement outcomes. Our goal is to find schemes that attain the optimal success probability for discrimination of measurements.

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