# 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

### 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.

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.

### ► References

[1] J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, Universal linear optics,'' Science, vol. 349, no. 6249, pp. 711–716, 2015.
https:/​/​doi.org/​10.1126/​science.aab3642

[2] S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Characterizing quantum supremacy in near-term devices,'' Nature Physics, vol. 14, no. 6, pp. 595–600, 2018.
https:/​/​doi.org/​10.1038/​s41567-018-0124-x

[3] J. Preskill, Quantum Computing in the NISQ era and beyond,'' Quantum, vol. 2, p. 79, 2018.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[4] E. Magesan, J. M. Gambetta, and J. Emerson, Scalable and Robust Randomized Benchmarking of Quantum Processes,'' Phyical Review Letters, vol. 106, p. 180504, 2011.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.180504

[5] L. Aolita, C. Gogolin, M. Kliesch, and J. Eisert, Reliable quantum certification of photonic state preparations,'' Nature Communications, vol. 6, p. 8498, 2015.
https:/​/​doi.org/​10.1038/​ncomms9498

[6] J. R. Wootton, Benchmarking of quantum processors with random circuits,'' arxiv:1806.02736, 2018.
arXiv:1806.02736

[7] J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, Quantum certification and benchmarking,'' Nature Reviews Physics, pp. 1–9, 2020.
https:/​/​doi.org/​10.1038/​s42254-020-0186-4

[8] A. Chefles, Quantum state discrimination,'' Contemporary Physics, vol. 41, no. 6, pp. 401–424, 2000.
https:/​/​doi.org/​10.1080/​00107510010002599

[9] S. M. Barnett and S. Croke, Quantum state discrimination,'' Advances in Optics and Photonics, vol. 1, no. 2, pp. 238–278, 2009.
https:/​/​doi.org/​10.1364/​AOP.1.000238

[10] J. A. Bergou, Discrimination of quantum states,'' Journal of Modern Optics, vol. 57, no. 3, pp. 160–180, 2010.
https:/​/​doi.org/​10.1080/​09500340903477756

[11] J. Bae and L.-C. Kwek, Quantum state discrimination and its applications,'' Journal of Physics A: Mathematical and General, vol. 48, no. 8, p. 083001, 2015.
https:/​/​doi.org/​10.1088/​1751-8113/​48/​8/​083001

[12] S. Pirandola, R. Laurenza, C. Lupo, and J. L. Pereira, Fundamental limits to quantum channel discrimination,'' npj Quantum Information, vol. 5, no. 1, pp. 1–8, 2019.
https:/​/​doi.org/​10.1038/​s41534-019-0162-y

[13] J. Watrous, The Theory of Quantum Information. Cambridge University Press, 2018.
https:/​/​doi.org/​10.1017/​9781316848142

[14] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, et al., Quantum supremacy using a programmable superconducting processor,'' Nature, vol. 574, no. 7779, pp. 505–510, 2019.
https:/​/​doi.org/​10.1038/​s41586-019-1666-5

[15] A. Acín, Statistical distinguishability between unitary operations,'' Physical Review Letters, vol. 87, no. 17, p. 177901, 2001.
https:/​/​doi.org/​10.1103/​PhysRevLett.87.177901

[16] R. Duan, Y. Feng, and M. Ying, Entanglement is not necessary for perfect discrimination between unitary operations,'' Physical Review Letters, vol. 98, no. 10, p. 100503, 2007.
https:/​/​doi.org/​10.1103/​PhysRevLett.98.100503

[17] R. Duan, Y. Feng, and M. Ying, Local distinguishability of multipartite unitary operations,'' Physical Review Letters, vol. 100, no. 2, p. 020503, 2008.
https:/​/​doi.org/​10.1103/​PhysRevLett.100.020503

[18] G. Chiribella, G. M. D'Ariano, and M. Roetteler, Identification of a reversible quantum gate: assessing the resources,'' New Journal of Physics, vol. 15, no. 10, p. 103019, 2013.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​10/​103019

[19] R. Duan, Y. Feng, and M. Ying, Perfect distinguishability of quantum operations,'' Physical Review Letters, vol. 103, no. 21, p. 210501, 2009.
https:/​/​doi.org/​10.1103/​PhysRevLett.103.210501

[20] A. W. Harrow, A. Hassidim, D. W. Leung, and J. Watrous, Adaptive versus nonadaptive strategies for quantum channel discrimination,'' Physical Review A, vol. 81, no. 3, p. 032339, 2010.
https:/​/​doi.org/​10.1103/​PhysRevA.81.032339

[21] T. P. Cope and S. Pirandola, Adaptive estimation and discrimination of Holevo-Werner channels,'' Quantum Measurements and Quantum Metrology, vol. 4, no. 1, pp. 44–52, 2017.
https:/​/​doi.org/​10.1515/​qmetro-2017-0006

[22] S. Pirandola and C. Lupo, Ultimate precision of adaptive noise estimation,'' Physical Review Letters, vol. 118, no. 10, p. 100502, 2017.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.100502

[23] A. Krawiec, Ł. Pawela, and Z. Puchała, Discrimination of POVMs with rank-one effects,'' Quantum Information Processing, vol. 19, no. 12, pp. 1–12, 2020.
https:/​/​doi.org/​10.1007/​s11128-020-02883-3

[24] R. Duan, C. Guo, C.-K. Li, and Y. Li, Parallel distinguishability of quantum operations,'' in 2016 IEEE International Symposium on Information Theory (ISIT), pp. 2259–2263, IEEE, 2016.
https:/​/​doi.org/​10.1109/​ISIT.2016.7541701

[25] I. Nechita, Z. Puchała, Ł. Pawela, and K. Życzkowski, Almost all quantum channels are equidistant,'' Journal of Mathematical Physics, vol. 59, no. 5, p. 052201, 2018.
https:/​/​doi.org/​10.1063/​1.5019322

[26] Z. Ji, Y. Feng, R. Duan, and M. Ying, Identification and distance measures of measurement apparatus,'' Physical Review Letters, vol. 96, no. 20, p. 200401, 2006.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.200401

[27] G. M. D'Ariano, P. L. Presti, and M. G. Paris, Using entanglement improves the precision of quantum measurements,'' Physical Review Letters, vol. 87, no. 27, p. 270404, 2001.
https:/​/​doi.org/​10.1103/​PhysRevLett.87.270404

[28] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Memory Effects in Quantum Channel Discrimination,'' Physical Review Letters, vol. 101, p. 180501, oct 2008.
https:/​/​doi.org/​10.1103/​PhysRevLett.101.180501

[29] Z. Puchała, Ł. Pawela, A. Krawiec, and R. Kukulski, Strategies for optimal single-shot discrimination of quantum measurements,'' Physical Review A, vol. 98, p. 042103, 2018.
https:/​/​doi.org/​10.1103/​PhysRevA.98.042103

[30] M. Sedlák and M. Ziman, Optimal single-shot strategies for discrimination of quantum measurements,'' Physical Review A, vol. 90, no. 5, p. 052312, 2014.
https:/​/​doi.org/​10.1103/​PhysRevA.90.052312

[31] D. Dieks, Overlap and distinguishability of quantum states,'' Physics Letters A, vol. 126, no. 5, pp. 303 – 306, 1988.
https:/​/​doi.org/​10.1016/​0375-9601(88)90840-7

[32] M. Hayashi, Discrimination of two channels by adaptive methods and its application to quantum system,'' IEEE Transactions on Information Theory, vol. 55, no. 8, pp. 3807–3820, 2009.
https:/​/​doi.org/​10.1109/​TIT.2009.2023726

[33] C. W. Helstrom, Quantum Detection and Estimation Theory. Elsevier, 1976.
https:/​/​doi.org/​10.1007/​BF01007479

[34] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Theoretical framework for quantum networks,'' Physical Review A, vol. 80, no. 2, p. 022339, 2009.
https:/​/​doi.org/​10.1103/​PhysRevA.80.022339

[35] A. Bisio, G. Chiribella, G. D'Ariano, and P. Perinotti, Quantum networks: general theory and applications,'' Acta Physica Slovaca. Reviews and Tutorials, vol. 61, no. 3, pp. 273–390, arxiv:1601.04864, 2011.
arXiv:1601.04864

[36] K. Korzekwa, S. Czachórski, Z. Puchała, and K. Życzkowski, Coherifying quantum channels,'' New Journal of Physics, vol. 20, no. 4, p. 043028, 2018.
https:/​/​doi.org/​10.1088/​1367-2630/​aaaff3

[37] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Memory effects in quantum channel discrimination,'' Physical Review Letters, vol. 101, no. 18, p. 180501, 2008.
https:/​/​doi.org/​10.1103/​PhysRevLett.101.180501

[38] I. Bengtsson and K. Życzkowski, Geometry of quantum states: an introduction to quantum entanglement. Cambridge University Press, 2017.
https:/​/​doi.org/​10.1017/​CBO9780511535048

[39] K. Życzkowski and H.-J. Sommers, Truncations of random unitary matrices,'' Journal of Physics A: Mathematical and General, vol. 33, no. 10, p. 2045, 2000.
https:/​/​doi.org/​10.1088/​0305-4470/​33/​10/​307

[40] R. Piziak, P. Odell, and R. Hahn, Constructing projections on sums and intersections,'' Computers $\&$ Mathematics with Applications, vol. 37, no. 1, pp. 67–74, 1999.
https:/​/​doi.org/​10.1016/​S0898-1221(98)00242-9

[41] M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín, Simulating Positive-Operator-Valued Measures with Projective Measurements,'' Physical Review Letters, vol. 119, p. 190501, Nov 2017.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.190501

[42] L. Guerini, J. Bavaresco, M. Terra Cunha, and A. Acín, Operational framework for quantum measurement simulability,'' Journal of Mathematical Physics, vol. 58, no. 9, p. 092102, 2017.
https:/​/​doi.org/​10.1063/​1.4994303

[43] F. Hausdorff, Der Wertvorrat einer Bilinearform,'' Mathematische Zeitschrift, vol. 3, no. 1, pp. 314–316, 1919.
https:/​/​doi.org/​10.1007/​BF01292610

[44] O. Toeplitz, Das algebraische Analogon zu einem Satze von Fejér,'' Mathematische Zeitschrift, vol. 2, no. 1-2, pp. 187–197, 1918.
https:/​/​doi.org/​10.1007/​BF01212904

[45] G. Jaeger and A. Shimony, Optimal distinction between two non-orthogonal quantum states,'' Physics Letters A, vol. 197, no. 2, pp. 83–87, 1995.
https:/​/​doi.org/​10.1016/​0375-9601(94)00919-G

### Cited by

[1] Filip B. Maciejewski, Zoltán Zimborás, and Michał Oszmaniec, "Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography", arXiv:1907.08518.

[2] Kamil Korzekwa, Stanisław Czachórski, Zbigniew Puchała, and Karol Życzkowski, "Distinguishing classically indistinguishable states and channels", Journal of Physics A Mathematical General 52 47, 475303 (2019).

[3] Aleksandra Krawiec, Łukasz Pawela, and Zbigniew Puchała, "Discrimination of POVMs with rank-one effects", Quantum Information Processing 19 12, 428 (2020).

[4] Kenji Nakahira and Kentaro Kato, "Ultimate limits to quantum process discrimination", arXiv:2012.13844.

[5] Paulina Lewandowska, Aleksandra Krawiec, Ryszard Kukulski, Łukasz Pawela, and Zbigniew Puchała, "On the optimal certification of von Neumann measurements", Scientific Reports 11, 3623 (2021).

[6] Chandan Datta, Tanmoy Biswas, Debashis Saha, and Remigiusz Augusiak, "Perfect discrimination of quantum measurements using entangled systems", arXiv:2012.07069.

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