Leif Ove Hansen

• Hansen, Leif Ove & Myrheim, Jan (2017). Nongeneric positive partial transpose states of rank five in 3 X 3 dimensions. International Journal of Quantum Information. ISSN 0219-7499. 15(7). doi: 10.1142/S021974991750054X.
• Hansen, Leif Ove; Hauge, Andreas; Myrheim, Jan & Sollid, Per Øyvind (2015). Extremal entanglement witnesses. International Journal of Quantum Information. ISSN 0219-7499. 13(08). doi: 10.1142/S0219749915500604.
• Hansen, Leif Ove & Myrheim, Jan (2015). Visualizing extremal positive maps in unital and trace-preserving form. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 92(4). doi: 10.1103/PhysRevA.92.042306.
• Hansen, Leif Ove; Hauge, Andreas; Myrheim, Jan & Sollid, Per Øyvind (2012). Low-rank positive-partial-transpose states and their relation to product vectors. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 85(2). doi: 10.1103/PhysRevA.85.022309. Vis sammendrag

  Deciding whether a mixed quantum state is separable or entangled is a difficult problem in general. Separable states are positive under partial transposition [they are positive-partial-transpose (PPT) states], but this simple test does not exclude all entangled states. In order to understand the entangled PPT states, having so-called bound entanglement, we want to study the extremal PPT states. An extremal PPT state is either a pure product state, then it is separable, or it is entangled, in which case the state and its partial transpose must still be rather low-rank density matrices, although it is known that the rank must be at least 4. In a previous paper we presented a complete classification of the rank-4 entangled PPT states in dimension 3×3, generalizing the construction by Bennett et al. from unextendible sets of orthogonal product vectors. In the present paper we continue our investigations of the low-rank entangled PPT states, mostly in dimension 3×3, using a combination of analytical and numerical methods. We use perturbation theory in order to construct rank-5 entangled PPT states close to the known rank-4 states, and in order to compute dimensions and study the geometry of surfaces of low-rank PPT states. We exploit the close connection between low-rank PPT states and product vectors. In particular, we show how to reconstruct a low-rank PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.

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