Phase transitions for random states and a semi-circle law for the partial transpose

Details Phase transitions for random states and a semi-circle law for the partial transpose Phase transitions for random states and a semi-circle law for the partial transpose

2011-12-20

arxiv.org/abs/1112.4582v2

Physical Review A (Rapid Communications) 85, 030302 (2012)

Physics

Quantum Physics

5 pages, 2 figures. This short note contains a high-level overview of two long and technical papers, arXiv:1011.0275 and arXiv

Scientific paper

10.1103/PhysRevA.85.030302

For a system of N identical particles in a random pure state, there is a threshold k_0 = k_0(N) ~ N/5 such that two subsystems of k particles each typically share entanglement if k > k_0, and typically do not share entanglement if k < k_0. By "random" we mean here "uniformly distributed on the sphere of the corresponding Hilbert space." The analogous phase transition for the positive partial transpose (PPT) property can be described even more precisely. For example, for N qubits the two subsystems of size k are typically in a PPT state if k < k_1 := N/4 - 1/2, and typically in a non-PPT state if k > k_1. Since, for a given state of the entire system, the induced state of a subsystem is given by the partial trace, the above facts can be rephrased as properties of random induced states. An important step in the analysis depends on identifying the asymptotic spectral density of the partial transposes of such random induced states, a result which is interesting in its own right.

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