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Title: Separability problem for multipartite states of rank at most 4
Authors: Chen, L. 
Doković, D.Z.
Issue Date: 12-Jul-2013
Citation: Chen, L., Doković, D.Z. (2013-07-12). Separability problem for multipartite states of rank at most 4. Journal of Physics A: Mathematical and Theoretical 46 (27) : -. ScholarBank@NUS Repository.
Abstract: 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⊗3 or a 2⊗2⊗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⊗3 or 2⊗2⊗2 system contains a product vector if and only if its Plücker 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. © 2013 IOP Publishing Ltd.
Source Title: Journal of Physics A: Mathematical and Theoretical
ISSN: 17518113
DOI: 10.1088/1751-8113/46/27/275304
Appears in Collections:Staff Publications

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