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Quantum Information Processing

Erschienen in:

01.10.2021

\(H_2\)-reducible matrices in six-dimensional mutually unbiased bases

verfasst von: Xiaoyu Chen, Mengfan Liang, Mengyao Hu, Lin Chen

Erschienen in: Quantum Information Processing | Ausgabe 10/2021

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Abstract

Finding four six-dimensional mutually unbiased bases (MUBs) containing the identity matrix is a long-standing open problem in quantum information. We show that if they exist, then the \(H_2\)-reducible matrix in the four MUBs has exactly nine \(2\times 2\) Hadamard submatrices. We apply our result to exclude from the four MUBs some known CHMs, such as symmetric \(H_2\)-reducible matrix, the Hermitian matrix, Dita family, Bjorck’s circulant matrix, and Szollosi family. Our results represent the latest progress on the existence of six-dimensional MUBs.

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Schwinger, J.: Unitary operator bases. Proc. Nat. Acad. Sci. U.S.A. 46(4), 570–579 (1960)ADSMathSciNetCrossRef

Amburg, I., Sharma, R., Sussman, D.M., Wootters, W.K.: States that “look the same” with respect to every basis in a mutually unbiased set. J. Math. Phys. 55(12), 122206 (2014)

Amburg, I., Sharma, R., Sussman, D.M., Wootters, W.K.: Erratum: “states that ‘look the same’ with respect to every basis in a mutually unbiased set”. J. Math. Phys. 56(3), 039901 (2015)

Brierley, S.: Mutually unbiased bases in low dimensions. University of York, Department of Mathematics. Ph.D. thesis (2009)

Brierley, S., Weigert, S.: Maximal sets of mutually unbiased quantum states in dimension 6. Phys. Rev. A 78, 042312 (2008)ADSMathSciNetCrossRef

Brierley, S., Weigert, S.: Constructing mutually unbiased bases in dimension six. Phys. Rev. A 79, 052316 (2009)ADSMathSciNetCrossRef

Brierley, S., Weigert, S.: Mutually unbiased bases and semi-definite programming. J. Phys. Conf. Ser. 254, 22 (2010)

Maxwell, A.S.: Brierley, Stephen: On properties of Karlsson Hadamards and sets of mutually unbiased bases in dimension six. Linear Algebra Appl. 466, 296–306 (2015)MathSciNetCrossRef

Szöllösi, F.: Complex hadamard matrices of order 6: a four-parameter family. Journal of the London Mathematical Society 85(3), 616–632 (2012)MathSciNetCrossRef

10.

Turek, O, Goyeneche D. A generalization of circulant Hadamard and conference matrices. Linear Algebra Appl. 569:241–265 (2016)MathSciNetCrossRef

11.

Nicoara, R.: Worley, Chase: A finiteness result for circulant core complex Hadamard matrices. Linear Algebra Appl. 571, 143–153 (2019)MathSciNetCrossRef

12.

Szöllősi, F.: Parametrizing complex hadamard matrices. Eur. J. Combin. 29(5), 1219–1234 (2008)MathSciNetCrossRef

13.

Goyeneche, D.: Mutually unbiased triplets from non-affine families of complex hadamard matrices in dimension 6. J. Phys. A: Math. Theor. 46(10), 105301 (2013)ADSMathSciNetCrossRef

14.

Boykin, P.O., Sitharam, M., Tiep, P.H., Wocjan, P.: Mutually unbiased bases and orthogonal decompositions of Lie algebras (2005). arXiv:quant-ph/0506089

15.

Jaming, P., Matolcsi, M., Móra, P., Szöllösi, F., Weiner, M.: A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6. J. Phys. Math. Theor. 42(24), 245305 (2009)ADSMathSciNetMATH

16.

Durt, T., Englert, B.-G., Bengtsson, I., Zyczkowski, K.: On mutually unbiased bases. Int. J. Quantum Information 8(4), 535–640 (2010)CrossRef

17.

Wiesniak, M., Paterek, T., Zeilinger, A.: Entanglement in mutually unbiased bases. New J. Phys. 13(5), 053047 (2011)ADSMathSciNetCrossRef

18.

Mcnulty, D., Weigert, S.: On the impossibility to extend triples of mutually unbiased product bases in dimension six. Int. J. Quant. Inform. 10(05), 125 (2012)MathSciNetCrossRef

19.

McNulty, D.: Weigert, Stefan: All mutually unbiased product bases in dimension 6. J. Phys. A: Math. Theor. 45(13), 135307 (2012)ADSMathSciNetCrossRef

20.

Raynal, P, Lü, X, Englert, B-G.: Mutually unbiased bases in six dimensions: The four most distant bases. Phys. Rev. A 83, 33 (2011)ADSCrossRef

21.

McNulty, D.: Weigert, Stefan: The limited role of mutually unbiased product bases in dimension 6. J. Phys. A: Math. Theor. 45(10), 102001 (2012)ADSMathSciNetCrossRef

22.

McNulty D, Pammer B, Weigert S: Mutually unbiased product bases for multiple qudits. J. Math. Phys. 57(3): 32 (2016)ADSMathSciNetCrossRef

23.

Chen, L, Li, Y.: Product states and Schmidt rank of mutually unbiased bases in dimension six. J. Phys. Math. General 50(47), 475304 (2017)ADSMathSciNet

24.

Chen, L, Li, Y.: Mutually unbiased bases in dimension six containing a product-vector basis. Quant. Inform. Process. 17(8), 198 (2018)ADSMathSciNetCrossRef

25.

Designolle, S, Skrzypczyk, P, Fröwis, F, Brunner, N.: Quantifying measurement incompatibility of mutually unbiased bases. Phys. Rev. Lett. 122: 504 (2018)CrossRef

26.

Karlsson, B.R.: Three-parameter complex hadamard matrices of order 6. Linear Algebra Appl. 434(1), 247–258 (2011)MathSciNetCrossRef

27.

Karlsson, B.R.: \({H}_2\)-reducible complex hadamard matrices of order 6. Linear Algebra Appl. 434(1), 239–246 (2011)MathSciNetCrossRef

28.

Liang, M, Mengyao, H, Chen, L, Chen, X: The \({H}_2\)-reducible matrix in four six-dimensional mutually unbiased bases. Quantum Inform. Process. 18(11), 352 (2019)ADSCrossRef

29.

De Baerdemacker, S, De Vos, A, Chen, L, Li, Y: The Birkhoff theorem for unitary matrices of arbitrary dimensions. Linear Algebra Appl 514, 151–164 (2017)MathSciNetCrossRef

30.

Chen, L.: Friedland, Shmuel: The tensor rank of tensor product of two three-qubit W states is eight. Linear Algebra Appl. 543, 1–16 (2018)MathSciNetCrossRef

31.

Wang, K, Chen, L, Shen, Y, Sun, Y, Zhao, L-J.: Constructing \(2\times 2\times 4\) and \(4\times 4\) unextendible product bases and positive-partial-transpose entangled states. Linear Multilinear Algebra 69:131–46 (2019)CrossRef

32.

Beauchamp, K., Nicoara, R.: Orthogonal maximal abelian \(*\)-subalgebras of the \(6\times 6\) matrices (2006). arXiv:math/0609076v1

Titel: -reducible matrices in six-dimensional mutually unbiased bases
verfasst von: Xiaoyu Chen
Mengfan Liang
Mengyao Hu
Lin Chen
Publikationsdatum: 01.10.2021
Verlag: Springer US
Erschienen in: Quantum Information Processing / Ausgabe 10/2021
Print ISSN: 1570-0755
Elektronische ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-021-03278-8

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