Facial structures of separable and PPT states

Tom 96 / 2011

Seung-Hyeok Kye Banach Center Publications 96 (2011), 257-269 MSC: Primary 81P15; Secondary 15A30, 46L05. DOI: 10.4064/bc96-0-17


A positive semi-definite block matrix (a state if it is normalized) is said to be separable if it is the sum of simple tensors of positive semi-definite matrices. A state is said to be entangled if it is not separable.

It is very difficult to detect the border between separable and entangled states. The PPT (positive partial transpose) criterion tells us that the partial transpose of a separable state is again positive semi-definite, as was observed by M. D. Choi in 1982 from the mathematics side.

In this expository note, we explain the facial structures of the cone of all PPT block matrices, which are naturally characterized by pairs of subspaces of (small) matrices. We also discuss which faces of PPT's induce faces of separables, and which faces of separables are induced by PPT's.


  • Seung-Hyeok KyeDepartment of Mathematics, Seoul National University
    Seoul 151-742, Korea

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