Multiple-shot and unambiguous discrimination of von Neumann measurements

Zbigniew Puchała; Łukasz Pawela; Aleksandra Krawiec; Ryszard Kukulski; Michał Oszmaniec

doi:10.22331/q-2021-04-06-425

Multiple-shot and unambiguous discrimination of von Neumann measurements

Zbigniew Puchała^1,2, Łukasz Pawela¹, Aleksandra Krawiec^1,3, Ryszard Kukulski^1,4, and Michał Oszmaniec^5,6

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

Published:	2021-04-06, volume 5, page 425
Eprint:	arXiv:1810.05122v4
Doi:	https://doi.org/10.22331/q-2021-04-06-425
Citation:	Quantum 5, 425 (2021).

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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 $\textit{in parallel}$ 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.

Featured image: Measurement discrimination schemes considered in our study: the parallel (left) and unambiguous (right) ones. The black boxes represent the unknown measurement used N times. The main difference between the schemes is that the latter can output a label stating that the scheme has failed. On the other hand, when we get an answer in the unambiguous scheme, we know for certain that the measurement associated with that label was present in the black box.

Popular summary

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.

► BibTeX data

@article{Puchala2021multipleshot,
  doi = {10.22331/q-2021-04-06-425},
  url = {https://doi.org/10.22331/q-2021-04-06-425},
  title = {Multiple-shot and unambiguous discrimination of von {N}eumann measurements},
  author = {Pucha{\l{}}a, Zbigniew and Pawela, {\L{}}ukasz and Krawiec, Aleksandra and Kukulski, Ryszard and Oszmaniec, Micha{\l{}}},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {425},
  month = apr,
  year = {2021}
}

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