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Random unitary matrices find a number of applications in quantum information science, and are central to the recently defined boson sampling algorithm for photons in linear optics. We describe an operationally simple method to directly implement Haar random unitary matrices in optical circuits, with no requirement for prior or explicit matrix calculations. Our physically-motivated and compact representation directly maps independent probability density functions for parameters in Haar random unitary matrices, to optical circuit components. We go on to extend the results to the case of random unitaries for qubits.
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We consider the empirical eigenvalue distribution of an $mtimes m$ principal submatrix of an $ntimes n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $frac{m}{n}=alpha$, the empirical spectral measure is well-approximated by a deterministic measure $mu_alpha$ supported on the unit disc. In earlier work, we showed that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $mu_alpha$ is typically of order $sqrt{frac{log(m)}{m}}$ or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.
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 to infty, by the Marchenko-Pastur distribution.
We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random band matrices and the problem of their Anderson transition. We finally describe a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.
Let $U$ be a Haar distributed matrix in $mathbb U(n)$ or $mathbb O (n)$. In a previous paper, we proved that after centering, the two-parameter process [T^{(n)} (s,t) = sum_{i leq lfloor ns rfloor, j leq lfloor ntrfloor} |U_{ij}|^2] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of $U$ by a random one, where each row (resp. column) is chosen with probability $s$ (resp. $t$) independently. We prove that the corresponding two-parameter process, after centering and normalization by $n^{-1/2}$ converges to a Gaussian process. On the way we meet other interesting convergences.
In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $operatorname{GL}_n(mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.
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