Criteria for Separability of Multipartite Quantum System
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Criteria for Separability of Multipartite Quantum System Yuanhong Tao · Weiwei Ding · Chang’e Li
Received: 12 August 2012 / Accepted: 20 October 2012 / Published online: 8 November 2012 © Springer Science+Business Media New York 2012
Abstract We first present the Hamel base of the density operator space for multipartite quantum system, and thus establish a representation of density matrix. Moreover, according to the structure of the density matrix for multipartite quantum system, we present two necessary criteria for separability of multipartite quantum system of arbitrary dimensions. Keywords Separability · Density matrix · Quantum entanglement
1 Introduction The quantum entanglement phenomenon is regarded as an important physical resource. It has played very important roles in quantum information processing such as quantum computation, quantum teleportation, dense coding, quantum cryptographic schemes and so on [1]. Although quantum entangled states are the key issue in quantum information, the theory about quantum entanglement is still far from complete. Until now there is not a uniform standard to determine whether a mixed state be in the entangled state or not. Recently, much effort has been devoted to finding criteria for separability [2–11]. In this note, we give a representation of density matrix for multipartite quantum system and present two necessary criteria for separability of multipartite quantum system of arbitrary dimensions. An independent R-dimensional Hilbert space of Hermitian operators can always be represented by identity operator I and the generators of the special unitary group SU (R). Generators of the group SU (R) can be constructed by R × R ordered elementary matrices {ejk | k, j = 1, 2, . . . , R}, where the ejk is the matrix whose numbers of kth row and j th column are one and the rest numbers are all zero [2]. There are R 2 − 1 typical
Supported by Natural Science Foundation of China (11161049); the Natural Science Foundation of Jilin Province (201215239). Y. Tao () · W. Ding · C. Li Department of Mathematics, Yanbian University, Jilin 133002, P.R. China e-mail: [email protected]
Int J Theor Phys (2013) 52:1970–1978
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generators of SU (R), which are all R × R ordered matrices with trace zero, denoted by {λi | i = 1, 2, . . . , R 2 − 1}. All these R 2 − 1 typical generators and the identity operator I construct a complete hermitian operator base of the space MR (C). Since every density operator of any quantum system is positive semi-definite Hermitian operator, the density operator can be represented by identity operator and the above typical generators of special unitary group SU (R). If ρ is the density matrix of R-dimensional single-particle quantum state, then it can be represented as 2 −1 R 1 2 IR + ci λi , ρ= 2 R i=1
(1)
2
R −1 where IR is R × R-order identity matrix, {λi }i=1 are the generators of SU (R), ci is the expectations of λi .
Lemma 1.1 [7] Let ρ be the density matrix of R-dimensional single-particle quantum state, which representation as (1). Then
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