Random Unitary Matrices Research Articles

Tail bounds for counts of zeros and eigenvalues, and an application to ratios

Let t be random and uniformly distributed in the interval [T,2T] , and consider the quantity N(t+1/\log\:T) - N(t) , a count of zeros of the Riemann zeta function in a box of height 1/\mathrm {log}\: T . Conditioned on the Riemann hypothesis, we show that the probability this count is greater than x decays at least as quickly as e^{-Cx\:\mathrm {log}\: x} , uniformly in T . We also prove a similar results for the logarithmic derivative of the zeta function, and likewise analogous results for the eigenvalues of a random unitary matrix. We use results of this sort to show on the Riemann hypothesis that the averages [ \frac{1}{T} \int_T^{2T} \Bigg| \frac{\zeta\big(\frac{1}{2} + \frac{\alpha}{\log T} + it\big)}{\zeta\big(\frac{1}{2}+ \frac{\beta}{\log T} + it\big)}\Bigg|^m,dt ] remain bounded as T\rightarrow\infty , for \alpha, \beta complex numbers with \beta\neq 0 . Moreover we show rigorously that the local distribution of zeros asymptotically controls ratio averages like the above; that is, the GUE Conjecture implies a (first-order) ratio conjecture.

Thinning and conditioning of the circular unitary ensemble

We apply the operation of random independent thinning on the eigenvalues of [Formula: see text] Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as [Formula: see text].

Direct dialling of Haar random unitary matrices

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.

Asymptotic behavior for log-determinants of several non-Hermitian random matrices

We study the asymptotic behavior for log-determinants of two unitary but non-Hermitian random matrices: the spherical ensembles A-1B; where A and B are independent complex Ginibre ensembles and the truncation of circular unitary ensembles. The corresponding Berry-Esseen bounds and Cramer type moderate deviations are established. Our method is based on the estimates of corresponding cumulants. Numerical simulations are also presented to illustrate the theoretical results.

Metric and classical fidelity uncertainty relations for random unitary matrices

We analyze uncertainty relations on finite dimensional Hilbert spaces expressed in terms of classical fidelity, which are stronger than metric uncertainty relations introduced by Fawzi, Hayden and Sen. We establish the validity of fidelity uncertainty relations for random unitary matrices with optimal parameters (up to universal constants) which improves upon known results for the weaker notion of metric uncertainty.This result is then applied to locking classical information in quantum states and allows to obtain optimal locking in Hellinger distance, improving upon previous results on locking in the total variation distance, both by strengthening the metric used and by improving the dependence on parameters.We also show that general probabilistic estimates behind the main theorem can be used to prove existence of data hiding schemes with Bayesian type guarantees.As a byproduct of our approach we obtain existence of almost Euclidean subspaces of the matrix spaces with a better dimension/distortion dependence than allowed in previously known constructions.

Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros

According to Dyson's threefold way, from the viewpoint of global time reversal symmetry, there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary, and symplectic ensembles, denoted COE, CUE, and CSE, respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability , we take up the problem of calculating the first two terms in the scaled large N expansion of the spacing distributions. It is well known that the leading term admits a characterization in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterizations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE, there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case, some further statistics are similarly analyzed.

Deflation as a Method of Variance Reduction for Estimating the Trace of a Matrix Inverse

Many fields require computing the trace of the inverse of a large, sparse matrix. The typical method used for such computations is the Hutchinson method which is a Monte Carlo (MC) averaging over matrix quadratures. To improve its convergence, several variance reductions techniques have been proposed. In this paper, we study the effects of deflating the near null singular value space. We make two main contributions. First, we analyze the variance of the Hutchinson method as a function of the deflated singular values and vectors. Although this provides good intuition in general, by assuming additionally that the singular vectors are random unitary matrices, we arrive at concise formulas for the deflated variance that include only the variance and mean of the singular values. We make the remarkable observation that deflation may increase variance for Hermitian matrices but not for non-Hermitian ones. This is a rare, if not unique, property where non-Hermitian matrices outperform Hermitian ones. The theory can be used as a model for predicting the benefits of deflation. Second, we use deflation in the context of a large scale application of "disconnected diagrams" in Lattice QCD. On lattices, Hierarchical Probing (HP) has previously provided an order of magnitude of variance reduction over MC by removing "error" from neighboring nodes of increasing distance in the lattice. Although deflation used directly on MC yields a limited improvement of 30% in our problem, when combined with HP they reduce variance by a factor of over 60 compared to MC. For this, we pre-computated 1000 smallest singular values of an ill-conditioned matrix of size 25 million. Using PRIMME and a domain-specific Algebraic Multigrid preconditioner, we perform one of the largest eigenvalue computations in Lattice QCD at a fraction of the cost of our trace computation.

Maximum of the Characteristic Polynomial of Random Unitary Matrices

It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a $N\times N$ random unitary matrix sampled from the Haar measure grows like $CN/(\log N)^{3/4}$ for some random variable $C$. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range $[N^{1 - \varepsilon},N^{1 + \varepsilon}]$, for arbitrarily small $\varepsilon$. The method is based on identifying an approximate branching random walk in the Fourier decomposition of the characteristic polynomial, and uses techniques developed to describe the extremes of branching random walks and of other log-correlated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimension-dependent symbols. The original argument for these asymptotics followed the general idea that the statistical mechanics of $1/f$-noise random energy models is governed by a freezing transition. We also prove the conjectured freezing of the free energy for random unitary matrices.

Self-similarity in the circular unitary ensemble

Self-similarity in the circular unitary ensemble, Discrete Analysis, 2016:9, 15 pp. The distribution of the eigenvalues of random matrices is a central topic of research. It includes famous results such as Wigner's semicircle law from 1955, which states that the distribution function of a large random symmetric matrix tends to a distribution whose density function, when suitably normalized, has a semicircular graph. (This result is true for several models of random symmetric matrices: in particular, it is true if all entries on and below the diagonal are independent Gaussians with mean 0 and variance 1, with the entries above the diagonal determined by symmetry.) It also received a huge boost in 1973 as a result of an observation by Hugh Montgomery and Freeman Dyson that the correlation between pairs of nearby zeros of the Riemann zeta function appears to be the same, after normalization, as the correlation between pairs of eigenvalues of a random unitary or Hermitian matrices. (Roughly speaking, the pair correlation tells you what you expect the distribution of the remaining points to be given where one point is. In both cases there is a repulsive effect: zeros of the zeta function do not like to be too close to each other, and neither do eigenvalues of these random matrices.) This paper concerns random unitary matrices. The eigenvalues of a unitary matrix lie on the unit circle in the complex plane, and the distribution is rotation invariant (since if one multiplies a random unitary matrix by a scalar of modulus 1, its distribution is unaffected). The main result of this paper is that it also satisfies a self-similarity property, which states, roughly, that the distribution of eigenvalues in an interval is similar to the distribution in the entire circle. To be more precise about it, suppose we take a random $nm\times nm$ unitary matrix, take the eigenvalues of that matrix that have arguments in the interval $[-\pi/m,\pi/m]$, and raise each one to the power $m$ (so that the interval $[-\pi/m,\pi/m]$ is stretched to the interval $[-\pi,\pi]$). Then the resulting distribution of points will closely resemble the distribution of eigenvalues of a random $n\times n$ unitary matrix. Closeness is measured in the $L_1$-_Wasserstein distance_, which is defined as follows. If $\mu$ and $\nu$ are two probability distributions on a metric space $M$, then their $L_1$-Wasserstein distance is the infimum of $\mathbb E\, d(X,Y)$ over all pairs of random variables $X$ and $Y$ such that $X$ has distribution $\mu$ and $Y$ has distribution $\nu$. Intuitively, it is the total amount by which you have to "move" $\mu$ to turn it into $\nu$ if you do so as economically as possible. This result proves a conjecture of Coram and Diaconis, in a slightly modified form. (In the original conjecture, one chooses not an interval of fixed length but an interval from one eigenvalue to the one $n$ places along as you go round the circle, and that interval is stretched and randomly rotated.) ^{_{[Article image](https://www.flickr.com/photos/swallowtailgardenseeds/15625678138/in/pool-publicdomain/) by Swallowtail Garden Seeds}}

The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios

We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called “microscopic” level, that is we consider the characteristic polynomial at points whose distance to 1 has order 1 / n. We prove that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. In order to deal with this issue, we couple all the dimensions n on a single probability space, in such a way that almost sure convergence occurs when n goes to infinity. The strong convergence results in this setup provide us with a new approach to ratios: we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form $$\exp (2 i \pi \alpha /n)$$ and related objects (such as the logarithmic derivative). We also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale. On the number theory side, inspired by the work by Keating and Snaith, we conjecture some new limit theorems for the value distribution of the Riemann zeta function on the critical line at the level of stochastic processes.

Improving Information Content in Compressed Sensing by Modifying the Random Re-Construction Matrices

Background/Objectives: Compressed Sensing (CS) is an efficient sensing paradigm which guarantees reasonable reconstruction with less number of samples. We aim to increase the reconstruction quality of signals in CS. Methods/ Statistical Analysis: The behavior of random matrices is analyzed and an efficient method for improving the reconstruction quality is developed in CS based ECG reconstruction applications. The method is compared against Biorthogonal wavelet based approaches. Findings: Our analysis reveals that introduction of a modified column vector in the reconstruction matrix, which contains the sum of all columns of random matrix increases the reconstruction quality in CS applications. This idea was applied to different sparsifying domains and the results are very encouraging. We studied the effect of doing this on the singular values and both unitary matrices U and V. The first singular value (Σ) shot up making the condition number high, however there was not much change in the other singular values. The matrix U seems to remain random unitary matrix, where as matrix V has one value becoming unity in its rank space. Application/Improvements: Compared to wavelet based approaches the method shows reasonable improvement in Percentage Root Square Deviation (PRD).

Local stability of the free additive convolution

We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3.

Multiparticle Correlations in Mesoscopic Scattering: Boson Sampling, Birthday Paradox, and Hong-Ou-Mandel Profiles.

The interplay between single-particle interference and quantum indistinguishability leads to signature correlations in many-body scattering. We uncover these with a semiclassical calculation of the transmission probabilities through mesoscopic cavities for systems of noninteracting particles. For chaotic cavities we provide the universal form of the first two moments of the transmission probabilities over ensembles of random unitary matrices, including weak localization and dephasing effects. If the incoming many-body state consists of two macroscopically occupied wave packets, their time delay drives a quantum-classical transition along a boundary determined by the bosonic birthday paradox. Mesoscopic chaotic scattering of Bose-Einstein condensates is, then, a realistic candidate to build a boson sampler and to observe the macroscopic Hong-Ou-Mandel effect.

Asymptotic entropic uncertainty relations

We analyze entropic uncertainty relations for two orthogonal measurements on a N-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are stronger than those obtained with use of the entropic uncertainty relation by Maassen and Uffink, and they are optimal up to additive constants. We also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations, which hold with high probability with respect to the random choice of bases. The lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by Wehner and Winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity. As a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the Haar measure, which are of independent interest.

Joint probability distributions for projection probabilities of random orthonormal states

The quantum chaos conjecture applied to a finite dimensional quantum system implies that such a system has eigenstates that show similar statistical properties as the column vectors of random orthogonal or unitary matrices. Here, we consider the different probabilities for obtaining a specific outcome in a projective measurement, provided the system is in one of its eigenstates. We then give analytic expressions for the joint probability density for these probabilities, with respect to the ensemble of random matrices. In the case of the unitary group, our results can be applied, also, to the phenomenon of universal conductance fluctuations, where the same mathematical quantities describe partial conductances in a two-terminal mesoscopic scattering problem with a finite number of modes in each terminal.

Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices

Based on the multivariate saddle point method we study the asymptotic behavior of the characteristic polynomials associated to Wishart type random matrices that are formed as products consisting of independent standard complex Gaussian and a truncated Haar distributed unitary random matrix. These polynomials form a general class of hypergeometric functions of type 2 F r . We describe the oscillatory behavior on the asymptotic interval of zeros by means of formulae of Plancherel–Rotach type and subsequently use it to obtain the limiting distribution of the suitably rescaled zeros. Moreover, we show that the asymptotic zero distribution lies in the class of Raney distributions and by introducing appropriate coordinates elementary and explicit characterizations are derived for the densities as well as for the distribution functions.

30 years of General Statistical Analysis and canonical equation K 60 for Hermitian matrices (A + B U C)(A + B U C)*, where U is a random unitary matrix

Abstract In this paper we apply the REFORM method for the deduction of the system of canonical equations K 60 for normalized spectral functions of the matrices [An + Bn Un (ε)Cn ][An + Bn Un (ε)Cn ]*.

Finite-size corrections in random matrix theory and Odlyzko’s dataset for the Riemann zeros

Odlyzko has computed a dataset listing more than 109successive Riemann zeros, starting from a zero number to beyond 1023. This dataset relates to random matrix theory as, according to the Montgomery–Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and co-workers, have usedN×Nrandom unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natural process to apply to the dataset is to minimize it by deleting each member independently with some specified probability, and we proceed to compute empirical two-point correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterize the order 1/N2correction term to the spacing distribution for random unitary matrices in terms of a second-order differential equation with coefficients that are Painlevé transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. In comparison to the Riemann zero data accurate agreement is exhibited.

Random Sampling of Quantum States: a Survey of Methods

The numerical generation of random quantum states (RQS) is an important procedure for investigations in quantum information science. Here we review some methods that may be used for performing that task. We start by presenting a simple procedure for generating random state vectors, for which the main tool is the random sampling of unbiased discrete probability distributions (DPD). Afterwards the creation of random density matrices is addressed. In this context we first present the standard method, which consists in using the spectral decomposition of a quantum state for getting RQS from random DPDs and random unitary matrices. In the sequence the Bloch vector parametrization method is described. This approach, despite being useful in several instances, is not in general convenient for RQS generation. In the last part of the article we regard the overparametrized method (OPM) and the related Ginibre and Bures techniques. The OPM can be used to create random positive semidefinite matrices with unit trace from randomly produced general complex matrices in a simple way that is friendly for numerical implementations. We consider a physically relevant issue related to the possible domains that may be used for the real and imaginary parts of the elements of such general complex matrices. Subsequently a too fast concentration of measure in the quantum state space that appears in this parametrization is noticed.

Random Matrix Ensembles with Singularities and a Hierarchy of Painlevé III Equations

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order k at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierarchy of higher order analogs to the PIII equation, which reduces to the PIII equation itself when the pole is simple.