Bound entangled states with extremal properties

Abstract

Following recent work of Beigi and Shor, we investigate PPT states that are "heavily entangled." We first exploit volumetric methods to show that in a randomly chosen direction, there are PPT states whose distance in trace norm from separable states is (asymptotically) at least 1/4. We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of 2-ε from the separable states, we need a dimension of 2^{poly(\log(1/ε))}, as opposed to 2^{poly(1/ε)} given by the construction of Beigi and Shor. We do so by exploiting the so called {\it private states}, introduced earlier in the context of quantum cryptography. We also provide a lower bound for the distance between private states and PPT states and investigate the distance between pure states and the set of PPT states.

8 pages, 2 figures

References in corpus (7)

• Quantum Communication With Zero-Capacity Channels
• The volume of separable states is super-doubly-exponentially small
• Entanglement thresholds for random induced states
• Noisy Preprocessing and the Distillation of Private States
• Unconditional privacy over channels which cannot convey quantum information
• Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
• Phase transitions for random states and a semi-circle law for the partial transpose

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• Construction and properties of a class of private states in arbitrary dimensions