Canonical Form and Separability of PPT States on Multiple Quantum Spaces
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چکیده
By using the “subtracting projectors” method in proving the separability of PPT states on multiple quantum spaces, we derive a canonical form of PPT states in C1 ⊗CK2 ⊗· · ·⊗CKm ⊗CN composite quantum systems with rank N , from which a sufficient separability condition for these states is presented. As the key resource in quantum information processing [1], quantum entanglement has resulted in the explosion of interest in quantum computing and communication in recent years [2]. The separability of quantum pure states is well understood [3]. However for mixed states, although there are already many results related to the separability criterion, the physical character and mathematical structure of the quantum entanglement are still far from being satisfied. The PPT (positive partial transpose) criteria play very important roles in the investigation of separability. It is of significance to investigate the general form of PPT states to study the separability and bound entangled states. In [4] the PPT (mixed) states on C ×C2 ×CN with rank N has been studied and a separability criterion has been obtained. These results have been generalized to the case of PPT states on C 2 × C × C with rank N [5], and the case of high dimensional tripartite systems [6] and C × C × C × C [7]. In this note we summarize our efforts in striving at some understanding of the properties of quantum entanglement for composite systems. We generalize the results to the most general case of the canonical form of PPT states in C K1 ⊗ C2 ⊗ · · · ⊗ Cm ⊗ C composite quantum systems with rank N , and study the separability condition for these states in terms of the canonical form. Let C be K-dimensional complex vector space with orthonormal basis {|i〉}, i = 0, ..., K − 1. A general pure state on C1 ⊗ C2 ⊗ . . .⊗ Cm is of the form |Ψ〉 = K1−1
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تاریخ انتشار 2005