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Journal of Scientific Computing

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01-04-2022

Rank-1 Approximation for Entangled Multipartite Real Systems

Authors: Matthew M. Lin, Moody T. Chu

Published in: Journal of Scientific Computing | Issue 1/2022

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Abstract

The interaction of multiple parts with each other within a system according to certain intrinsic rules is a crucial natural phenomenon. The notion of entanglement and its decomposition of high-dimensional arrays is particularly intriguing since it opens a new way of thinking in data processing and communication, of which the applications will be broad and significant. Depending on how the internal parts engage with each other, there are different types of entanglements with distinct characteristics. This paper concerns the approximation over a multipartite system whose subsystems consist of symmetric rank-1 matrices that are entangled via the Kronecker tensor product. Such a structure resembles that arising in quantum mechanics where a mixed state is to be approximated by its nearest separable state, except that the discussion in this paper is limited to real-valued matrices. Unlike the conventional low-rank tensor approximations, the added twist due to the involvement of the Kronecker product destroys the multi-linearity, which makes the problem harder. As a first step, this paper explores the rank-1 multipartite approximation only. Reformulated as a nonlinear eigenvalue problem and a nonlinear singular value problem, respectively, the problem can be tackled numerically by power-like iterative methods and SVD-like iterative methods. The iteration in both classes of methods can be implemented cyclically or acyclically. Motivations, schemes, and convergence theory are discussed in this paper. Preliminary numerical experiments suggest these methods are effective and efficient when compared with some general-purpose optimization packages.

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Aaronson, S.: Quantum Computing Since Democritus. Cambridge University Press, Cambridge (2013)CrossRef

Ben-Aryeh, Y., Mann, A.: Biseparability of 3-qubits density matrices using Hilbert–Schmidt decompositions: sufficient conditions and explicit expressions. Int. J. Quantum Inf. 14(08), 1650044 (2016). https://doi.org/10.1142/S0219749916500441MathSciNetCrossRefMATH

Brachat, J., Comon, P., Mourrain, B., Tsigaridas, E.: Symmetric tensor decomposition. Linear Algebra Appl. 433(11–12), 1851–1872 (2010). https://doi.org/10.1016/j.laa.2010.06.046MathSciNetCrossRefMATH

Braunstein, S.L., Caves, C.M., Jozsa, R., Linden, N., Popescu, S., Schack, R.: Separability of very noisy mixed states and implications for NMR quantum computing. Phys. Rev. Lett. 83, 1054–1057 (1999). https://doi.org/10.1103/PhysRevLett.83.1054CrossRef

Chen, K., Wu, L.A.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. 3(3), 193–202 (2003)MathSciNetMATH

Chen, L., Aulbach, M., Hajdušek, M.: Comparison of different definitions of the geometric measure of entanglement. Phys. Rev. A 89, 042305 (2014). https://doi.org/10.1103/PhysRevA.89.042305CrossRef

Chu, M.T., Lin, M.M.: Nonlinear power-like and SVD-like iterative schemes with applications to entangled bipartite rank-1 approximation. SIAM J. Sci. Comput. 43(5), S448–S474 (2021). https://doi.org/10.1137/20M1336059MathSciNetCrossRefMATH

Comon, P., Luciani, X., de Almeida, A.L.F.: Tensor decompositions, alternating least squares and other tales. J. Chemom. 23, 393–405 (2009). https://doi.org/10.1002/cem.1236CrossRef

Dana, M., Ikramov, K.D.: On the codimension of the variety of symmetric matrices with multiple eigenvalues. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 323(Chisl. Metody i Vopr. Organ. Vychisl. 18), 34–46, 224 (2005). https://doi.org/10.1007/s10958-006-0275-7

10.

De Lathauwer, L., De Moor, B., Vandewalle, J.: On the best rank-1 and rank-\((R_1, R_2,\ldots , R_N)\) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000). https://doi.org/10.1137/S0895479898346995MathSciNetCrossRefMATH

11.

Dür, W., Cirac, J.I.: Classification of multiqubit mixed states: separability and distillability properties. Phys. Rev. A 61, 042314 (2000). https://doi.org/10.1103/PhysRevA.61.042314MathSciNetCrossRef

12.

Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A (3) 62(6), 062314 (2000). https://doi.org/10.1103/PhysRevA.62.062314MathSciNetCrossRef

13.

Eltschka, C., Siewert, J.: Entanglement of three-qubit Greenberger–Horne–Zeilinger-symmetric states. Phys. Rev. Lett. 108, 020502 (2012). https://doi.org/10.1103/PhysRevLett.108.020502CrossRef

14.

García, C.B., Li, T.Y.: On the number of solutions to polynomial systems of equations. SIAM J. Numer. Anal. 17(4), 540–546 (1980). https://doi.org/10.1137/0717046MathSciNetCrossRefMATH

15.

Gharibian, S.: Strong np-hardness of the quantum separability problem. Quantum Inf. Comput. 10(3), 343–360 (2010)MathSciNetMATH

16.

Goncharov, A.V., Hrkac, G., Dean, J.S., Schrefl, T.: Kronecker product approximation of demagnetizing tensors for micromagnetics. J. Comput. Phys. 229(7), 2544–2549 (2010). https://doi.org/10.1016/j.jcp.2009.12.004MathSciNetCrossRefMATH

17.

Gorbachev, V.N., Trubilko, A.I., Rodichkina, A.A., Zhiliba, A.I.: Can the states of the W-class be suitable for teleportation? Phys. Lett. A 314(4), 267–271 (2003). https://doi.org/10.1016/S0375-9601(03)00906-XMathSciNetCrossRefMATH

18.

Grasedyck, L.: Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing 72, 247–265 (2004). https://doi.org/10.1007/s00607-003-0037-zMathSciNetCrossRefMATH

19.

Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem (2007)

20.

Guan, Y., Chu, M.T., Chu, D.: Convergence analysis of an SVD-based algorithm for the best rank-1 tensor approximation. Linear Algebra Appl. 555, 53–69 (2018). https://doi.org/10.1016/j.laa.2018.06.006MathSciNetCrossRefMATH

21.

Gurvits, L.: Classical complexity and quantum entanglement. J. Comput. Syst. Sci. 69(3), 448–484 (2004)MathSciNetCrossRef

22.

Hackbusch, W., Khoromskij, B.N., Tyrtyshnikov, E.E.: Hierarchical Kronecker tensor-product approximations. J. Numer. Math. 13(2), 119–156 (2005). https://doi.org/10.1163/1569395054012767MathSciNetCrossRefMATH

23.

Hiai, F., Petz, D.: Introduction to Matrix Analysis and Applications. Universitext. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04150-6CrossRefMATH

24.

Ishteva, M., Absil, P.A., Van Dooren, P.: Jacobi algorithm for the best low multilinear rank approximation of symmetric tensors. SIAM J. Matrix Anal. Appl. 34(2), 651–672 (2013). https://doi.org/10.1137/11085743XMathSciNetCrossRefMATH

25.

Joo, J., Park, Y.J., Oh, S., Kim, J.: Quantum teleportation via a W state. New J. Phys. 5, 136–136 (2003). https://doi.org/10.1088/1367-2630/5/1/136CrossRef

26.

Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23(3), 863–884 (2001). https://doi.org/10.1137/S0895479801387413MathSciNetCrossRefMATH

27.

Kolda, T.G.: Numerical optimization for symmetric tensor decomposition. Math. Program. 151(1, Ser. B), 225–248 (2015). https://doi.org/10.1007/s10107-015-0895-0MathSciNetCrossRefMATH

28.

Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009). https://doi.org/10.1137/07070111XMathSciNetCrossRefMATH

29.

Langville, A.N., Stewart, W.J.: A Kronecker product approximate preconditioner for SANs. Numer. Linear Algebra Appl. 11(8–9), 723–752 (2004). https://doi.org/10.1002/nla.344MathSciNetCrossRefMATH

30.

Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4(3), 553–572 (1983). https://doi.org/10.1137/0904038MathSciNetCrossRefMATH

31.

Murao, M., Plenio, M.B., Popescu, S., Vedral, V., Knight, P.L.: Multiparticle entanglement purification protocols. Phys. Rev. A 57, R4075–R4078 (1998). https://doi.org/10.1103/PhysRevA.57.R4075CrossRef

32.

Nagy, J., Kilmer, M.: Kronecker product approximation for preconditioning in three-dimensional imaging applications. IEEE Trans. Image Process. 15(3), 604–613 (2006)MathSciNetCrossRef

33.

Nakahara, M., Ohmi, T.: Quantum Computing. From Linear Algebra to Physical Realizations. CRC Press, Boca Raton (2008). https://doi.org/10.1201/9781420012293CrossRefMATH

34.

Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, Cambridge (2010)CrossRef

35.

Rezghi, M., Hosseini, S.M., Eldén, L.: Best Kronecker product approximation of the blurring operator in three dimensional image restoration problems. SIAM J. Matrix Anal. Appl. 35(3), 1086–1104 (2014). https://doi.org/10.1137/130917260MathSciNetCrossRefMATH

36.

Silvestre, D., Hespanha, J., Silvestre, C.: A pagerank algorithm based on asynchronous Gauss-Seidel iterations. In: 2018 Annual American Control Conference (ACC), pp. 484–489 (2018)

37.

Sommese, A.J., Wampler, C.W., II.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005). https://doi.org/10.1142/9789812567727CrossRefMATH

38.

Uschmajew, A.: Local convergence of the alternating least squares algorithm for canonical tensor approximation. SIAM J. Matrix Anal. Appl. 33(2), 639–652 (2012). https://doi.org/10.1137/110843587MathSciNetCrossRefMATH

39.

Van Loan, C.F.: The ubiquitous Kronecker product. J. Comput. Appl. Math. 123(1–2), 85–100 (2000). https://doi.org/10.1016/S0377-0427(00)00393-9MathSciNetCrossRefMATH

40.

Van Loan, C.F., Pitsianis, N.: Approximation with Kronecker products. In: Linear Algebra for Large Scale and Real-Time Applications (Leuven, 1992), NATO Adv. Sci. Inst. Ser. E Appl. Sci., vol. 232, pp. 293–314. Kluwer Acad. Publ., Dordrecht (1993)

41.

Vervliet, N., Debals, O., Sorber, L., Van Barel, M., De Lathauwer, L.: Tensorlab 3.0. https://www.tensorlab.net (2016)

42.

Vidal, G., Tarrach, R.: Robustness of entanglement. Phys. Rev. A 59, 141–155 (1999). https://doi.org/10.1103/PhysRevA.59.141MathSciNetCrossRef

43.

Wang, L., Chu, M.T.: On the global convergence of the alternating least squares method for rank-one approximation to generic tensors. SIAM J. Matrix Anal. Appl. 35(3), 1058–1072 (2014). https://doi.org/10.1137/130938207MathSciNetCrossRefMATH

44.

Zhang, T., Golub, G.H.: Rank-one approximation to high order tensors. SIAM J. Matrix Anal. Appl. 23(2), 534–550 (2001). https://doi.org/10.1137/S0895479899352045MathSciNetCrossRefMATH

45.

Zhao, X.L., Huang, T.Z., Lv, X.G., Xu, Z.B., Huang, J.: Kronecker product approximations for image restoration with new mean boundary conditions. Appl. Math. Model. 36(1), 225–237 (2012). https://doi.org/10.1016/j.apm.2011.05.050MathSciNetCrossRefMATH

Title: Rank-1 Approximation for Entangled Multipartite Real Systems
Authors: Matthew M. Lin
Moody T. Chu
Publication date: 01-04-2022
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2022
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-022-01805-y

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