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Quantum marginals: the generic entanglement regime

Monday 14th October 2013 - 15:30 to 16:30
INI Seminar Room 1
Co-authors: Stanislaw SZAREK (Universite Paris 6 & Case Western Reserve University), Deping YE (Memorial University of Newfoundland)

Consider N qubits in a generic pure state on $(C^2)^{\otimes N}$. Give a fraction $p \in (0,1/2)$ of them to Alice, another fraction $p$ of them to Bob, and assume the remaining qubits disappear in the environment. Do Alice and Bob share some entanglement ?

When N is large, this problem features a threshold phenomenon around the critical value $p=1/5$. In this talk I give an elementary proof of the "easy" half of this result: for $p>1/5$, entanglement is generic. The general question, and a discussion of the threshold phenomenon, will be addressed in the talk by S. Szarek.

I will introduce background from high-dimensional convex geometry and prove some key estimates on the size (specifically the mean width) of the set of separable states.

The talk is a variation on arxiv:1106.2264 (joint with S. Szarek and D. Ye).

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons