Abstract
We study entanglement-related properties of random quantum states which are unitarily invariant, in the sense that their distribution is left unchanged by conjugation with arbitrary unitary operators. In the large matrix size limit, the distribution of these random quantum states is characterized by their limiting spectrum, a compactly supported probability distribution. We prove several results characterizing entanglement and the PPT property of random bipartite unitarily invariant quantum states in terms of the limiting spectral distribution, in the unbalanced asymptotical regime where one of the two subsystems is fixed, while the other one grows in size.
Highlights
In quantum information theory, when one needs to understand properties of typical density matrices, it is necessary to endow the convex body of quantum states with a natural, physically motivated probability measure, in order to compute statistics of the relevant quantities
The induced measures, introduced by Zyczkowski and Sommers [2], but already studied by Page [1], have received the most attention, mainly due to their simplicity and to their natural physical interpretation: a density matrix from the induced ensemble is obtained by tracing an environment system of appropriate dimension out of a random uniform bipartite pure state
In [5], Aubrun studied bipartite random quantum states from the induced ensemble, and determined which values of the ratio environment size/system size the random states are, with high probability, PPT
Summary
In quantum information theory, when one needs to understand properties of typical density matrices, it is necessary to endow the convex body of quantum states with a natural, physically motivated probability measure, in order to compute statistics of the relevant quantities. We consider random quantum states which have the property that their distribution is left unchanged by conjugation with arbitrary unitary operations; we call them unitarily invariant These distributions are characterized only by their spectrum, and we consider sequences of distributions with the property that their spectra converge towards some compactly supported probability measure μ on the real line. The family of distributions we consider generalizes the induced ensemble, which corresponds to a Marcenko-Pastur limiting spectral distribution We provide conditions such that the quantum state corresponding to a random unitarily invariant matrix will be, with large probability, PPT, separable, or entangled.
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