Entropy of entanglement and multifractal exponents for random states

Physical Review A (Impact Factor: 2.99). 08/2008; DOI: 10.1103/PhysRevA.79.032308
Source: arXiv

ABSTRACT We relate the entropy of entanglement of ensembles of random vectors to their generalized fractal dimensions. Expanding the von Neumann entropy around its maximum we show that the first order only depends on the participation ratio, while higher orders involve other multifractal exponents. These results can be applied to entanglement behavior near the Anderson transition.

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    Physical Review A 02/2012; 85(6). · 2.99 Impact Factor
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    ABSTRACT: We investigate numerically the position- and momentum-space Shannon information entropies, S_{x}^{β} and S_{p}^{β}, respectively, of energy eigenstates |β〉 for an electron in four kinds of one-dimensional (1D) nonuniform systems, i.e., the Harper model, the slowly varying potential ones, the complex quasiperiodic potential ones, and the random-dimer potential ones. In the former three models, electronic localization properties are well-defined. For them, we find it interesting that, S_{x}^{β} is greater than, equal to, and less than S_{p}^{β} for delocalized, critical, and localized states in position-space, respectively, which can be used as signatures of the transition from a delocalized phase to a localized ones. With the criterion, we analyze the random-dimer potential model. We give another perspective and propose a consistent interpretation of discrepancies about the random-dimer potential model. Therefore, all these provide us a simple method to discern the nature of states in these 1D nonuniform systems.
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    ABSTRACT: We compute analytically the statistics of the Renyi and von Neumann entropies (standard measures of entanglement), for a random pure state in a large bipartite quantum system. The full probability distribution is computed by first mapping the problem to a random matrix model and then using a Coulomb gas method. We identify three different regimes in the entropy distribution, which correspond to two phase transitions in the associated Coulomb gas. The two critical points correspond to sudden changes in the shape of the Coulomb charge density: the appearance of an integrable singularity at the origin for the first critical point, and the detachement of the rightmost charge (largest eigenvalue) from the sea of the other charges at the second critical point. Analytical results are verified by Monte Carlo numerical simulations. A short account of some of these results appeared recently in Phys. Rev. Lett. {\bf 104}, 110501 (2010).
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