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arXiv:1006.0088 · Journal of Physics A Mathematical and Theoretical

Entangled random pure states with orthogonal symmetry: exact results

Pierpaolo Vivo

Published Sep 15, 2010• 16 citations
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Abstract

We compute analytically the density ϱ N , M (λ) of Schmidt eigenvalues, distributed according to a fixed-trace Wishart–Laguerre measure, and the average Rényi entropy for reduced density matrices of entangled random pure states with orthogonal symmetry (β = 1). The results are valid for arbitrary dimensions N = 2 k , M of the corresponding Hilbert space partitions, and are in excellent agreement with numerical simulations.

DOI
10.1088/1751-8113/43/40/405206

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We compute analytically the density ϱ N , M (λ) of Schmidt eigenvalues, distributed according to a fixed-trace Wishart–Laguerre measure, and the average Rényi entropy for reduced density matrices of entangled random pure states with orthogonal symmetry (β = 1). The results are valid for arbitrary dimensions N = 2 k , M of the corresponding Hilbert space partitions, and are in excellent agreement with numerical simulations.
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The entanglement spectrum (i.e., the full distribution of Schmidt eigenvalues of the reduced density matrix) contains more information than the conventional entanglement entropy and has been studied recently in several many-particle systems. We compute the disorder-averaged entanglement spectrum in the form of the disorder-averaged moments $\stackrel{\underline{}}{\mathrm{Tr}{\mathrm{\ensuremath{\rho}}}_{\mathrm{A}}^{\mathrm{\ensuremath{\alpha}}}}$ of the reduced density matrix ${\ensuremath{\rho}}_{A}$ for a contiguous block of many spins at the random-singlet quantum critical point in one dimension. The result compares well in the scaling limit with numerical studies on the random XX\mathit{XX}XX model and is also expected to describe the (interacting) random Heisenberg model. Our numerical studies on the XX\mathit{XX}XX case reveal that the dependence of the entanglement entropy and spectrum on the geometry of the Hilbert space partition is quite different than for conformally invariant critical points.
We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.
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