A universal scheme for robust self-testing in the prepare-and-measure scenario
1Institute of Theoretical Physics and Astrophysics, National Quantum Information Center, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-306 Gdańsk, Poland
2International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdańsk, Poland
3Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
| Published: | 2021-04-06, volume 5, page 424 |
| Eprint: | arXiv:2003.01032v5 |
| Doi: | https://doi.org/10.22331/q-2021-04-06-424 |
| Citation: | Quantum 5, 424 (2021). |
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Abstract
We consider the problem of certification of arbitrary ensembles of pure states and projective measurements solely from the experimental statistics in the prepare-and-measure scenario assuming the upper bound on the dimension of the Hilbert space. To this aim, we propose a universal and intuitive scheme based on establishing perfect correlations between target states and suitably-chosen projective measurements. The method works in all finite dimensions and allows for robust certification of the overlaps between arbitrary preparation states and between the corresponding measurement operators. Finally, we prove that for qubits, our technique can be used to robustly self-test arbitrary configurations of pure quantum states and projective measurements. These results pave the way towards the practical application of the prepare-and-measure paradigm to certification of quantum devices.
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[1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Land huis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
https://doi.org/10.1038/s41586-019-1666-5
[2] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Reviews of Modern Physics 89, 035002 (2017).
https://doi.org/10.1103/RevModPhys.89.035002
[3] I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Reviews of Modern Physics 86, 153 (2014).
https://doi.org/10.1103/RevModPhys.86.153
[4] A. Montanaro, Quantum algorithms: an overview, npj Quantum Information 2, 1 (2016).
https://doi.org/10.1038/npjqi.2015.23
[5] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature 549, 195 (2017).
https://doi.org/10.1038/nature23474
[6] V. Dunjko and H. J. Briegel, Machine learning & artificial intelligence in the quantum domain: a review of recent progress, Reports on Progress in Physics 81, 074001 (2018).
https://doi.org/10.1088/1361-6633/aab406
[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 2, 382 (2020).
https://doi.org/10.1038/s42254-020-0186-4
[8] D. Mayers and A. Yao, Self testing quantum apparatus, arXiv preprint quant-ph/0307205 (2003).
arXiv:quant-ph/0307205
[9] I. Šupić and J. Bowles, Self-testing of quantum systems: a review, Quantum 4, 337 (2020).
https://doi.org/10.22331/q-2020-09-30-337
[10] J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1, 195 (1964).
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
[11] S.-L. Chen, C. Budroni, Y.-C. Liang, and Y.-N. Chen, Natural framework for device-independent quantification of quantum steerability, measurement incompatibility, and self-testing, Physical Review Letters 116, 240401 (2016).
https://doi.org/10.1103/PhysRevLett.116.240401
[12] A. Coladangelo, K. T. Goh, and V. Scarani, All pure bipartite entangled states can be self-tested, Nature Communications 8, 15485 (2017).
https://doi.org/10.1038/ncomms15485
[13] J. Bowles, I. Šupić, D. Cavalcanti, and A. Acín, Device-independent entanglement certification of all entangled states, Physical Review Letters 121, 180503 (2018).
https://doi.org/10.1103/PhysRevLett.121.180503
[14] S. Popescu and D. Rohrlich, Which states violate Bell's inequality maximally? Physics Letters A 169, 411 (1992).
https://doi.org/10.1016/0375-9601(92)90819-8
[15] J. Ahrens, P. Badziąg, M. Pawłowski, M. Żukowski, and M. Bourennane, Experimental tests of classical and quantum dimensionality, Physical Review Letters 112, 140401 (2014).
https://doi.org/10.1103/PhysRevLett.112.140401
[16] J. B. Brask, A. Martin, W. Esposito, R. Houlmann, J. Bowles, H. Zbinden, and N. Brunner, Megahertz-rate semi-device-independent quantum random number generators based on unambiguous state discrimination, Physical Review Applied 7, 054018 (2017).
https://doi.org/10.1103/PhysRevApplied.7.054018
[17] E. A. Aguilar, M. Farkas, D. Martínez, M. Alvarado, J. Cariñe, G. B. Xavier, J. F. Barra, G. Cañas, M. Pawłowski, and G. Lima, Certifying an irreducible 1024-dimensional photonic state using refined dimension witnesses, Physical Review Letters 120, 230503 (2018).
https://doi.org/10.1103/PhysRevLett.120.230503
[18] H. Anwer, S. Muhammad, W. Cherifi, N. Miklin, A. Tavakoli, and M. Bourennane, Experimental characterization of unsharp qubit observables and sequential measurement incompatibility via quantum random access codes, Physical Review Letters 125, 080403 (2020).
https://doi.org/10.1103/PhysRevLett.125.080403
[19] M. Pawłowski and N. Brunner, Semi-device-independent security of one-way quantum key distribution, Physical Review A 84, 010302 (2011).
https://doi.org/10.1103/PhysRevA.84.010302
[20] T. Van Himbeeck, E. Woodhead, N. J. Cerf, R. García-Patrón, and S. Pironio, Semi-device-independent framework based on natural physical assumptions, Quantum 1, 33 (2017).
https://doi.org/10.22331/q-2017-11-18-33
[21] R. Chaves, J. B. Brask, and N. Brunner, Device-independent tests of entropy, Physical Review Letters 115, 110501 (2015).
https://doi.org/10.1103/PhysRevLett.115.110501
[22] T. Fritz, Quantum correlations in the temporal Clauser–Horne–Shimony–Holt (CHSH) scenario, New Journal of Physics 12, 083055 (2010).
https://doi.org/10.1088/1367-2630/12/8/083055
[23] A. Tavakoli, J. Kaniewski, T. Vértesi, D. Rosset, and N. Brunner, Self-testing quantum states and measurements in the prepare-and-measure scenario, Physical Review A 98, 062307 (2018).
https://doi.org/10.1103/PhysRevA.98.062307
[24] M. Farkas and J. Kaniewski, Self-testing mutually unbiased bases in the prepare-and-measure scenario, Physical Review A 99, 032316 (2019).
https://doi.org/10.1103/PhysRevA.99.032316
[25] P. Mironowicz and M. Pawłowski, Experimentally feasible semi-device-independent certification of four-outcome positive-operator-valued measurements, Physical Review A 100, 030301 (2019).
https://doi.org/10.1103/PhysRevA.100.030301
[26] A. Tavakoli, M. Smania, T. Vértesi, N. Brunner, and M. Bourennane, Self-testing non-projective quantum measurements, Science Advances 6, eaaw6664 (2020).
https://doi.org/10.1126/sciadv.aaw6664
[27] A. Tavakoli, D. Rosset, and M.-O. Renou, Enabling computation of correlation bounds for finite-dimensional quantum systems via symmetrization, Physical Review Letters 122, 070501 (2019).
https://doi.org/10.1103/PhysRevLett.122.070501
[28] R. Blume-Kohout, J. K. Gamble, E. Nielsen, K. Rudinger, J. Mizrahi, K. Fortier, and P. Maunz, Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography, Nature Communications 8, 1 (2017).
https://doi.org/10.1038/ncomms14485
[29] C. H. Bennett and G. Brassard, Quantum cryptography: Public-key distribution and coin tossing, Theoretical Computer Science 560, 7 (2014).
https://doi.org/10.1016/j.tcs.2014.05.025
[30] C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Physical Review Letters 68, 3121 (1992).
https://doi.org/10.1103/PhysRevLett.68.3121
[31] E. Woodhead, C. C. W. Lim, and S. Pironio, Semi-device-independent QKD based on BB84 and a CHSH-type estimation, In Theory of Quantum Computation, Communication, and Cryptography, vol 7582, pages 107–115, Springer, Berlin, Heidelberg (2013).
https://doi.org/10.1007/978-3-642-35656-8_9
[32] J. I. de Vicente, Shared randomness and device-independent dimension witnessing, Physical Review A 95, 012340 (2017).
https://doi.org/10.1103/PhysRevA.95.012340
[33] M. Navascués and S. Popescu, How energy conservation limits our measurements, Physical Review Letters 112, 140502 (2014).
https://doi.org/10.1103/PhysRevLett.112.140502
[34] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric informationally complete quantum measurements, Journal of Mathematical Physics 45, 2171 (2004).
https://doi.org/10.1063/1.1737053
[35] S. Brierley, S. Weigert, and I. Bengtsson, All mutually unbiased bases in dimensions two to five, arXiv preprint arXiv:0907.4097 (2009).
arXiv:0907.4097
[36] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge University Press, USA, 10th edition (2011).
https://doi.org/10.1017/CBO9780511976667
[37] V. Bargmann, Note on Wigner's theorem on symmetry operations, Journal of Mathematical Physics 5, 862 (1964).
https://doi.org/10.1063/1.1704188
[38] J.-G. Sun, Perturbation bounds for the Cholesky and QR factorizations, BIT Numerical Mathematics 31, 341 (1991).
https://doi.org/10.1007/BF01931293
[39] E. Arias-Castro, A. Javanmard, and B. Pelletier, Perturbation bounds for Procrustes, classical scaling, and trilateration, with applications to manifold learning, Journal of Machine Learning Research 21, 15 (2020).
http://jmlr.org/papers/v21/18-720.html
[40] H. Bechmann-Pasquinucci and N. Gisin, Intermediate states in quantum cryptography and Bell inequalities, Physical Review A 67, 062310 (2003).
https://doi.org/10.1103/PhysRevA.67.062310
[41] G. M. D'Ariano, P. L. Presti, and P. Perinotti, Classical randomness in quantum measurements, Journal of Physics A: Mathematical and General 38, 5979 (2005).
https://doi.org/10.1088/0305-4470/38/26/010
[42] M. Oszmaniec and T. Biswas, Operational relevance of resource theories of quantum measurements, Quantum 3, 133 (2019).
https://doi.org/10.22331/q-2019-04-26-133
[43] R. Takagi and B. Regula, General resource theories in quantum mechanics and beyond: operational characterization via discrimination tasks, Physical Review X 9, 031053 (2019).
https://doi.org/10.1103/PhysRevX.9.031053
[44] R. Uola, T. Kraft, J. Shang, X.-D. Yu, and O. Gühne, Quantifying quantum resources with conic programming, Physical Review Letters 122, 130404 (2019).
https://doi.org/10.1103/PhysRevLett.122.130404
[45] A. Tavakoli, Semi-device-independent certification of independent quantum state and measurement devices, Physical Review Letters 125, 150503 (2020).
https://doi.org/10.1103/PhysRevLett.125.150503
[46] N. J. Higham, Accuracy and stability of numerical algorithms, Society for Industrial and Applied Mathematics, USA, 2nd edition (2002).
https://doi.org/10.1137/1.9780898718027
[47] X.-W. Chang and D. Stehlé, Rigorous perturbation bounds of some matrix factorizations, SIAM Journal on Matrix Analysis and Applications 31, 2841 (2010).
https://doi.org/10.1137/090778535
[48] J. R. Hurley and R. B. Cattell, The Procrustes program: Producing direct rotation to test a hypothesized factor structure, Behavioral Science 7, 258 (1962).
https://doi.org/10.1002/bs.3830070216
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