Perfect Embezzlement of Entangled States

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Hayden and Van Dam showed that starting with a separable state in Alice and Bob’s state space and a shared entangled state in a common bipartite resource space, then using local unitary operations, it is possible to produce an entangled pair in the state space while at the same time only perturbing the shared entangled state by a small amount, which can be made arbitrarily small as the dimension of the resource space grows. They referred to this as “embezzling entanglement” since numerically it “appears" that the resource state was returned exactly.

It is natural to wonder if using an infinite dimensional resource space and local operations, one can return the resource state exactly while producing an entangled state in their state space. Whether or not you can achieve this phenomenon of “perfect embezzlement of an entangled state” depends on which mathematical model one uses to describe “local”.

We prove that perfect embezzlement is impossible in the tensor model but is possible in the commuting model. We then relate this to current work on the conjectures of Connes and Tsirelson about different models for quantum conditional probabilities.

This talk is based on joint work with R. Cleve, L. Liu and S. Harris.