Separability problem for multipartite states of rank at most 4

被引:34
作者
Chen, Lin [1 ,2 ,3 ]
Dokovic, Dragomir Z. [1 ,2 ]
机构
[1] Univ Waterloo, Dept Pure Math, Waterloo, ON N2L 3G1, Canada
[2] Univ Waterloo, Inst Quantum Comp, Waterloo, ON N2L 3G1, Canada
[3] Natl Univ Singapore, Ctr Quantum Technol, Singapore 117542, Singapore
基金
加拿大自然科学与工程研究理事会;
关键词
UNEXTENDIBLE PRODUCT BASES; ENTANGLED STATES; QUANTUM STATES;
D O I
10.1088/1751-8113/46/27/275304
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
One of the most important problems in quantum information is the separability problem, which asks whether a given quantum state is separable. We investigate multipartite states of rank at most 4 which are PPT (i.e., all their partial transposes are positive semidefinite). We show that any PPT state of rank 2 or 3 is separable and has length at most 4. For separable states of rank 4, we show that they have length at most 6. It is six only for some qubit-qutrit or multiqubit states. It turns out that any PPT entangled state of rank 4 is necessarily supported on a 3 circle times 3 or a 2 circle times 2 circle times 2 subsystem. We obtain a very simple criterion for the separability problem of the PPT states of rank at most 4: such a state is entangled if and only if its range contains no product vectors. This criterion can be easily applied since a four-dimensional subspace in the 3 circle times 3 or 2 circle times 2 circle times 2 system contains a product vector if and only if its Plucker coordinates satisfy a homogeneous polynomial equation (the Chow form of the corresponding Segre variety). We have computed an explicit determinantal expression for the Chow form in the former case, while such an expression was already known in the latter case.
引用
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页数:24
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