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

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• Bipartite Bound Entanglement
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• Iterative optimization in quantum metrology and entanglement theory using semidefinite programming
• Bound entanglement-assisted prepare-and-measure scenarios based on four-dimensional quantum messages
• Bound entanglement in symmetric random induced states
• A Class of PPT Entangled States Arbitrary Far From Separable States
• Construction and properties of a class of private states in arbitrary dimensions