Frontiers of Mathematics in China >
A successive approximation method for quantum separability
Received date: 14 Jul 2011
Accepted date: 12 Dec 2012
Published date: 01 Dec 2013
Copyright
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.
Key words： Quantum system; entanglement; tensor; successive approximation; M-eigenvalue; cross-hill
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
|
| 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
|
| 4 |
Doherty A C, Parrilo P A, Spedalieri F M. Distinguishing separable and entangled states. Phys. Rev. Lett, 2002, 88: 187904
|
| 5 |
Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete? Phys Rev, 1935, 47: 777
|
| 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
|
| 8 |
Han D, Qi L, Wu Ed. Extreme diffusion values for non-Gaussian diffusions. Optim Methods Softw, 2008, 23: 703-716
|
| 9 |
Horodecki M, Horodecki P, Horodecki R. Separability of mixed states: Necessary and sufficient conditions. Phys Lett A, 1996, 223: 1-8
|
| 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
|
| 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
|
| 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
|
| 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
|
| 15 |
Pittenger A O, Rubin M H. Geometry of entanglement witness and local detection of entanglement. Phys Rev A, 2003, 67: 012327
|
| 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
|
| 17 |
Qi L. Eigenvalues and invariants of tensors. J Math Anal Appl, 2007, 325: 1363-1377
|
| 18 |
Qi L, Dai H, Han D. Conditions for strong ellipticity. Front Math China, 2009, 4: 349-364
|
| 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
|
| 21 |
Qi L, Wang Y, Wu Ed. D-eigenvalues of diffusion kurtosis tensors. J Comp Appl Math, 2008, 221: 150-157
|
| 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
|
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