Worst-case Quantum Hypothesis Testing with Separable Measurements

Le Phuc Thinh1,2, Michele Dall'Arno1,3,4, and Valerio Scarani1,5

1Centre for Quantum Technologies, National University of Singapore, Singapore
2Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany
3Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyoku, Kyoto 606-8502, Japan
4Faculty of Education and Integrated Arts and Sciences, Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo 169-8050, Japan
5Department of Physics, National University of Singapore, Singapore

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

For any pair of quantum states (the hypotheses), the task of binary quantum hypotheses testing is to derive the tradeoff relation between the probability $p_{01}$ of rejecting the null hypothesis and $p_{10}$ of accepting the alternative hypothesis. The case when both hypotheses are explicitly given was solved in the pioneering work by Helstrom. Here, instead, for any given null hypothesis as a pure state, we consider the worst-case alternative hypothesis that maximizes $p_{10}$ under a constraint on the distinguishability of such hypotheses. Additionally, we restrict the optimization to separable measurements, in order to describe tests that are performed locally. The case $p_{01}=0$ has been recently studied under the name of ``quantum state verification''. We show that the problem can be cast as a semi-definite program (SDP). Then we study in detail the two-qubit case. A comprehensive study in parameter space is done by solving the SDP numerically. We also obtain analytical solutions in the case of commuting hypotheses, and in the case where the two hypotheses can be orthogonal (in the latter case, we prove that the restriction to separable measurements generically prevents perfect distinguishability). In regards to quantum state verification, our work shows the existence of more efficient strategies for noisy measurement scenarios.

► BibTeX data

► References

[1] Holevo A. Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam, 1st edition, 1982. 10.1007/​978-88-7642-378-9. URL https:/​/​doi.org/​10.1007/​978-88-7642-378-9.
https:/​/​doi.org/​10.1007/​978-88-7642-378-9

[2] A. Acín. Statistical distinguishability between unitary operations. Phys. Rev. Lett., 87: 177901, Oct 2001. 10.1103/​PhysRevLett.87.177901. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.87.177901.
https:/​/​doi.org/​10.1103/​PhysRevLett.87.177901

[3] Joonwoo Bae and Leong-Chuan Kwek. Quantum state discrimination and its applications. Journal of Physics A: Mathematical and Theoretical, 48 (8): 083001, jan 2015. 10.1088/​1751-8113/​48/​8/​083001. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1751-8113/​48/​8/​083001

[4] Alessandro Bisio, Michele Dall'Arno, and Giacomo Mauro D'Ariano. Tradeoff between energy and error in the discrimination of quantum-optical devices. Phys. Rev. A, 84: 012310, 2011. 10.1103/​PhysRevA.84.012310.
https:/​/​doi.org/​10.1103/​PhysRevA.84.012310

[5] Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. 10.1017/​CBO9780511804441.
https:/​/​doi.org/​10.1017/​CBO9780511804441

[6] J. Calsamiglia, J. I. de Vicente, R. Muñoz Tapia, and E. Bagan. Local discrimination of mixed states. Phys. Rev. Lett., 105: 080504, Aug 2010. 10.1103/​PhysRevLett.105.080504. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.105.080504.
https:/​/​doi.org/​10.1103/​PhysRevLett.105.080504

[7] Andrew M. Childs, John Preskill, and Joseph Renes. Quantum information and precision measurement. Journal of Modern Optics, 47 (2-3): 155–176, 2000. 10.1080/​09500340008244034. URL https:/​/​www.tandfonline.com/​doi/​abs/​10.1080/​09500340008244034.
https:/​/​doi.org/​10.1080/​09500340008244034

[8] Sarah Croke, Erika Andersson, Stephen M. Barnett, Claire R. Gilson, and John Jeffers. Maximum confidence quantum measurements. Phys. Rev. Lett., 96: 070401, Feb 2006. 10.1103/​PhysRevLett.96.070401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.96.070401.
https:/​/​doi.org/​10.1103/​PhysRevLett.96.070401

[9] Michele Dall'Arno, Alessandro Bisio, Giacomo Mauro D'Ariano, Martina Mikova, Miroslav Jezek, and Miloslav Dusek. Experimental implementation of unambiguous quantum reading. Phys. Rev. A, 85: 012308, 2012. 10.1103/​PhysRevA.85.012308.
https:/​/​doi.org/​10.1103/​PhysRevA.85.012308

[10] A. C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. Distinguishing separable and entangled states. Phys. Rev. Lett., 88: 187904, Apr 2002. 10.1103/​PhysRevLett.88.187904. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.88.187904.
https:/​/​doi.org/​10.1103/​PhysRevLett.88.187904

[11] Aram W. Harrow, Anand Natarajan, and Xiaodi Wu. An improved semidefinite programming hierarchy for testing entanglement. Communications in Mathematical Physics, 352 (3): 881–904, Jun 2017. ISSN 1432-0916. 10.1007/​s00220-017-2859-0. URL https:/​/​doi.org/​10.1007/​s00220-017-2859-0.
https:/​/​doi.org/​10.1007/​s00220-017-2859-0

[12] A. Hayashi, T. Hashimoto, and M. Horibe. State discrimination with error margin and its locality. Phys. Rev. A, 78: 012333, Jul 2008. 10.1103/​PhysRevA.78.012333. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.78.012333.
https:/​/​doi.org/​10.1103/​PhysRevA.78.012333

[13] M. Hayashi and H. Nagaoka. General formulas for capacity of classical-quantum channels. IEEE Transactions on Information Theory, 49 (7): 1753–1768, July 2003. 10.1109/​TIT.2003.813556.
https:/​/​doi.org/​10.1109/​TIT.2003.813556

[14] Masahito Hayashi. Group theoretical study of LOCC-detection of maximally entangled states using hypothesis testing. New Journal of Physics, 11 (4): 043028, apr 2009. 10.1088/​1367-2630/​11/​4/​043028. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​1367-2630/​11/​4/​043028

[15] Masahito Hayashi, Keiji Matsumoto, and Yoshiyuki Tsuda. A study of LOCC-detection of a maximally entangled state using hypothesis testing. Journal of Physics A: Mathematical and General, 39 (46): 14427–14446, nov 2006. 10.1088/​0305-4470/​39/​46/​013. URL https:/​/​doi.org/​10.1088.
https:/​/​doi.org/​10.1088/​0305-4470/​39/​46/​013

[16] Carl W Helstrom. Quantum detection and estimation theory. Journal of Statistical Physics, 1 (2): 231–252, 1969. 10.1007/​BF01007479.
https:/​/​doi.org/​10.1007/​BF01007479

[17] Z. Hradil. Quantum-state estimation. Phys. Rev. A, 55: R1561–R1564, Mar 1997. 10.1103/​PhysRevA.55.R1561. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.55.R1561.
https:/​/​doi.org/​10.1103/​PhysRevA.55.R1561

[18] Ye-Chao Liu, Xiao-Dong Yu, Jiangwei Shang, Huangjun Zhu, and Xiangdong Zhang. Efficient verification of dicke states. Phys. Rev. Applied, 12: 044020, Oct 2019. 10.1103/​PhysRevApplied.12.044020. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevApplied.12.044020.
https:/​/​doi.org/​10.1103/​PhysRevApplied.12.044020

[19] Seth Lloyd. Enhanced sensitivity of photodetection via quantum illumination. Science, 321 (5895): 1463–1465, 2008. ISSN 0036-8075. 10.1126/​science.1160627. URL https:/​/​science.sciencemag.org/​content/​321/​5895/​1463.
https:/​/​doi.org/​10.1126/​science.1160627
https:/​/​science.sciencemag.org/​content/​321/​5895/​1463

[20] Sam Pallister, Noah Linden, and Ashley Montanaro. Optimal verification of entangled states with local measurements. Phys. Rev. Lett., 120: 170502, Apr 2018. 10.1103/​PhysRevLett.120.170502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.120.170502.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.170502

[21] Matteo Paris and Jaroslav Rehacek. Quantum State Estimation. Springer Publishing Company, Incorporated, 1st edition, 2010. ISBN 3642061036, 9783642061035. 10.1007/​b98673.
https:/​/​doi.org/​10.1007/​b98673

[22] Stefano Pirandola. Quantum reading of a classical digital memory. Phys. Rev. Lett., 106: 090504, Mar 2011. 10.1103/​PhysRevLett.106.090504. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.106.090504.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.090504

[23] Yuki Takeuchi and Tomoyuki Morimae. Verification of many-qubit states. Phys. Rev. X, 8: 021060, Jun 2018. 10.1103/​PhysRevX.8.021060. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.8.021060.
https:/​/​doi.org/​10.1103/​PhysRevX.8.021060

[24] Kun Wang and Masahito Hayashi. Optimal verification of two-qubit pure states. Phys. Rev. A, 100: 032315, Sep 2019. 10.1103/​PhysRevA.100.032315. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.100.032315.
https:/​/​doi.org/​10.1103/​PhysRevA.100.032315

[25] Xiao-Dong Yu, Jiangwei Shang, and Otfried Gühne. Optimal verification of general bipartite pure states. npj Quantum Information, 5 (1): 112, 2019. ISSN 2056-6387. 10.1038/​s41534-019-0226-z. URL https:/​/​doi.org/​10.1038/​s41534-019-0226-z.
https:/​/​doi.org/​10.1038/​s41534-019-0226-z

[26] Huangjun Zhu and Masahito Hayashi. Optimal verification and fidelity estimation of maximally entangled states. Phys. Rev. A, 99: 052346, May 2019a. 10.1103/​PhysRevA.99.052346. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.99.052346.
https:/​/​doi.org/​10.1103/​PhysRevA.99.052346

[27] Huangjun Zhu and Masahito Hayashi. Efficient verification of hypergraph states. Phys. Rev. Applied, 12: 054047, Nov 2019b. 10.1103/​PhysRevApplied.12.054047. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevApplied.12.054047.
https:/​/​doi.org/​10.1103/​PhysRevApplied.12.054047

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2020-11-25 08:57:47). On SAO/NASA ADS no data on citing works was found (last attempt 2020-11-25 08:57:47).