RESEARCH ARTICLE

A successive approximation method for quantum separability

  • Deren HAN , 1 ,
  • Liqun QI 2
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  • 1. School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China
  • 2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

Received date: 14 Jul 2011

Accepted date: 12 Dec 2012

Published date: 01 Dec 2013

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

Determining whether a quantum state is separable or inseparable (entangled) is a problem of fundamental importance in quantum science and has attracted much attention since its first recognition by Einstein, Podolsky and Rosen [Phys. Rev., 1935, 47: 777] and Schröodinger [Naturwissenschaften, 1935, 23: 807-812, 823-828, 844-849]. In this paper, we propose a successive approximation method (SAM) for this problem, which approximates a given quantum state by a so-called separable state: if the given states is separable, this method finds its rank-one components and the associated weights; otherwise, this method finds the distance between the given state to the set of separable states, which gives information about the degree of entanglement in the system. The key task per iteration is to find a feasible descent direction, which is equivalent to finding the largest M-eigenvalue of a fourth-order tensor. We give a direct method for this problem when the dimension of the tensor is 2 and a heuristic cross-hill method for cases of high dimension. Some numerical results and experiences are presented.

Cite this article

Deren HAN , Liqun QI . A successive approximation method for quantum separability[J]. Frontiers of Mathematics in China, 2013 , 8(6) : 1275 -1293 . DOI: 10.1007/s11464-013-0274-1

1
Bazarra M S, Sherali H D, Shetty C M. Nonlinear Programming: Theory and Algorithms. New York: John Wiley and Sons, Inc, 1993

2
Cox D, Little J, O’Shea D. Using Algebraic Geometry. New York: Springer-Verlag, 1998

DOI

3
Dahl D, Leinass J M, Myrheim J, Ovrum E. A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl, 2007, 420: 711-725

DOI

4
Doherty A C, Parrilo P A, Spedalieri F M. Distinguishing separable and entangled states. Phys. Rev. Lett, 2002, 88: 187904

DOI

5
Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete? Phys Rev, 1935, 47: 777

DOI

6
Gurvits L. Classical deterministic complexity of Edmonds’ problem and quantum entanglement. IN: Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing. New York: ACM, 2003, 10-19

7
Han D, Dai H, Qi L. Conditions for strong ellipticity of anisotropic elastic materials. J Elasticity, 2009, 97: 1-13

DOI

8
Han D, Qi L, Wu Ed. Extreme diffusion values for non-Gaussian diffusions. Optim Methods Softw, 2008, 23: 703-716

DOI

9
Horodecki M, Horodecki P, Horodecki R. Separability of mixed states: Necessary and sufficient conditions. Phys Lett A, 1996, 223: 1-8

DOI

10
Ioannou L M, Travaglione B C, Cheung D, Ekert K. Improved algorithm for quantum separability and entanglement detection. Phys Rev A, 2004, 70: 060303

DOI

11
Ling C, Nie J, Qi L, Ye Y. Bi-quadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J Optim, 2009, 20: 1286-1310

DOI

12
Ng M, Qi L, Zhou G. Finding the largest eigenvalue of a non-negative tensor. SIAM J Matrix Anal Appl, 2009, 31: 1090-1099

DOI

13
Nielsen M N, Chuang I L. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000

14
Peres A. Separability criterion for density matrices. Phys Rev Lett, 1996, 77: 1413-1415

DOI

15
Pittenger A O, Rubin M H. Geometry of entanglement witness and local detection of entanglement. Phys Rev A, 2003, 67: 012327

DOI

16
Qi L. Rank and eigenvalues of a supersymmetric tensor, a multivariate homogeneous polynomial and an algebraic surface defined by them. J Symbolic Comput, 2006, 41: 1309-1327

DOI

17
Qi L. Eigenvalues and invariants of tensors. J Math Anal Appl, 2007, 325: 1363-1377

DOI

18
Qi L, Dai H, Han D. Conditions for strong ellipticity. Front Math China, 2009, 4: 349-364

DOI

19
Qi L, Wang F, Wang Y. A global homogenous polynomial optimization problem over the unit sphere. Working Paper, Department of Applied Mathematics, The Hong Kong Polytechnic University, 2007

20
Qi L, Wang F, Wang Y. Z-eigenvalue methods for a global polynomial optimization problem. Math Program, 2009, 118: 301-316

DOI

21
Qi L, Wang Y, Wu Ed. D-eigenvalues of diffusion kurtosis tensors. J Comp Appl Math, 2008, 221: 150-157

DOI

22
Schrödinger E. Die gegenwärtige situation in der quantenmechanik. Naturwissenschaften, 1935, 23: 807-812, 823-828, 844-849

23
Verstraete F, Dehaene J, De Moor B. On the geometry of entangled states. J Mod Opt, 2002, 49: 1277-1287

DOI

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