60 research outputs found

    Random matrix approach in search for weak signals immersed in background noise

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    We present new, original and alternative method for searching signals coded in noisy data. The method is based on the properties of random matrix eigenvalue spectra. First, we describe general ideas and support them with results of numerical simulations for basic periodic signals immersed in artificial stochastic noise. Then, the main effort is put to examine the strength of a new method in investigation of data content taken from the real astrophysical NAUTILUS detector, searching for the presence of gravitational waves. Our method discovers some previously unknown problems with data aggregation in this experiment. We provide also the results of new method applied to the entire respond signal from ground based detectors in future experimental activities with reduced background noise level. We indicate good performance of our method what makes it a positive predictor for further applications in many areas.Comment: 15 pages, 16 figure

    Number statistics for β\beta-ensembles of random matrices: applications to trapped fermions at zero temperature

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    Let Pβ(V)(NI)\mathcal{P}_{\beta}^{(V)} (N_{\cal I}) be the probability that a N×NN\times N β\beta-ensemble of random matrices with confining potential V(x)V(x) has NIN_{\cal I} eigenvalues inside an interval I=[a,b]{\cal I}=[a,b] of the real line. We introduce a general formalism, based on the Coulomb gas technique and the resolvent method, to compute analytically Pβ(V)(NI)\mathcal{P}_{\beta}^{(V)} (N_{\cal I}) for large NN. We show that this probability scales for large NN as Pβ(V)(NI)exp(βN2ψ(V)(NI/N))\mathcal{P}_{\beta}^{(V)} (N_{\cal I})\approx \exp\left(-\beta N^2 \psi^{(V)}(N_{\cal I} /N)\right), where β\beta is the Dyson index of the ensemble. The rate function ψ(V)(kI)\psi^{(V)}(k_{\cal I}), independent of β\beta, is computed in terms of single integrals that can be easily evaluated numerically. The general formalism is then applied to the classical β\beta-Gaussian (I=[L,L]{\cal I}=[-L,L]), β\beta-Wishart (I=[1,L]{\cal I}=[1,L]) and β\beta-Cauchy (I=[L,L]{\cal I}=[-L,L]) ensembles. Expanding the rate function around its minimum, we find that generically the number variance Var(NI){\rm Var}(N_{\cal I}) exhibits a non-monotonic behavior as a function of the size of the interval, with a maximum that can be precisely characterized. These analytical results, corroborated by numerical simulations, provide the full counting statistics of many systems where random matrix models apply. In particular, we present results for the full counting statistics of zero temperature one-dimensional spinless fermions in a harmonic trap.Comment: 34 pages, 19 figure

    Controlling Light Through Optical Disordered Media : Transmission Matrix Approach

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    We experimentally measure the monochromatic transmission matrix (TM) of an optical multiple scattering medium using a spatial light modulator together with a phase-shifting interferometry measurement method. The TM contains all information needed to shape the scattered output field at will or to detect an image through the medium. We confront theory and experiment for these applications and we study the effect of noise on the reconstruction method. We also extracted from the TM informations about the statistical properties of the medium and the light transport whitin it. In particular, we are able to isolate the contributions of the Memory Effect (ME) and measure its attenuation length

    Detecting entanglement of random states with an entanglement witness

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    The entanglement content of high-dimensional random pure states is almost maximal, nevertheless, we show that, due to the complexity of such states, the detection of their entanglement using witness operators is rather difficult. We discuss the case of unknown random states, and the case of known random states for which we can optimize the entanglement witness. Moreover, we show that coarse graining, modeled by considering mixtures of m random states instead of pure ones, leads to a decay in the entanglement detection probability exponential with m. Our results also allow to explain the emergence of classicality in coarse grained quantum chaotic dynamics.Comment: 14 pages, 4 figures; minor typos correcte

    Spectra of Empirical Auto-Covariance Matrices

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    We compute spectra of sample auto-covariance matrices of second order stationary stochastic processes. We look at a limit in which both the matrix dimension NN and the sample size MM used to define empirical averages diverge, with their ratio α=N/M\alpha=N/M kept fixed. We find a remarkable scaling relation which expresses the spectral density ρ(λ)\rho(\lambda) of sample auto-covariance matrices for processes with dynamical correlations as a continuous superposition of appropriately rescaled copies of the spectral density ρα(0)(λ)\rho^{(0)}_\alpha(\lambda) for a sequence of uncorrelated random variables. The rescaling factors are given by the Fourier transform C^(q)\hat C(q) of the auto-covariance function of the stochastic process. We also obtain a closed-form approximation for the scaling function ρα(0)(λ)\rho^{(0)}_\alpha(\lambda). This depends on the shape parameter α\alpha, but is otherwise universal: it is independent of the details of the underlying random variables, provided only they have finite variance. Our results are corroborated by numerical simulations using auto-regressive processes.Comment: 4 pages, 2 figure

    Rates of convergence for empirical spectral measures: a soft approach

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    Understanding the limiting behavior of eigenvalues of random matrices is the central problem of random matrix theory. Classical limit results are known for many models, and there has been significant recent progress in obtaining more quantitative, non-asymptotic results. In this paper, we describe a systematic approach to bounding rates of convergence and proving tail inequalities for the empirical spectral measures of a wide variety of random matrix ensembles. We illustrate the approach by proving asymptotically almost sure rates of convergence of the empirical spectral measure in the following ensembles: Wigner matrices, Wishart matrices, Haar-distributed matrices from the compact classical groups, powers of Haar matrices, randomized sums and random compressions of Hermitian matrices, a random matrix model for the Hamiltonians of quantum spin glasses, and finally the complex Ginibre ensemble. Many of the results appeared previously and are being collected and described here as illustrations of the general method; however, some details (particularly in the Wigner and Wishart cases) are new. Our approach makes use of techniques from probability in Banach spaces, in particular concentration of measure and bounds for suprema of stochastic processes, in combination with more classical tools from matrix analysis, approximation theory, and Fourier analysis. It is highly flexible, as evidenced by the broad list of examples. It is moreover based largely on "soft" methods, and involves little hard analysis

    Correlators for the Wigner–Smith time-delay matrix of chaotic cavities

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    We study the Wigner–Smith time-delay matrix Q of a ballistic quantum dot supporting N scattering channels. We compute the v-point correlators of the power traces Tr Qk for arbitrary v>1 at leading order for large N using techniques from the random matrix theory approach to quantum chromodynamics. We conjecture that the cumulants of the Tr Qkʼs are integer-valued at leading order in N and include a MATHEMATICA code that computes their generating functions recursively

    Subsystem dynamics under random Hamiltonian evolution

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    We study time evolution of a subsystem's density matrix under unitary evolution, generated by a sufficiently complex, say quantum chaotic, Hamiltonian, modeled by a random matrix. We exactly calculate all coherences, purity and fluctuations. We show that the reduced density matrix can be described in terms of a noncentral correlated Wishart ensemble for which we are able to perform analytical calculations of the eigenvalue density. Our description accounts for a transition from an arbitrary initial state towards a random state at large times, enabling us to determine the convergence time after which random states are reached. We identify and describe a number of other interesting features, like a series of collisions between the largest eigenvalue and the bulk, accompanied by a phase transition in its distribution function.Comment: 16 pages, 8 figures; v3: slightly re-structured and an additional appendi

    Practical recipes for the model order reduction, dynamical simulation, and compressive sampling of large-scale open quantum systems

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    This article presents numerical recipes for simulating high-temperature and non-equilibrium quantum spin systems that are continuously measured and controlled. The notion of a spin system is broadly conceived, in order to encompass macroscopic test masses as the limiting case of large-j spins. The simulation technique has three stages: first the deliberate introduction of noise into the simulation, then the conversion of that noise into an equivalent continuous measurement and control process, and finally, projection of the trajectory onto a state-space manifold having reduced dimensionality and possessing a Kahler potential of multi-linear form. The resulting simulation formalism is used to construct a positive P-representation for the thermal density matrix. Single-spin detection by magnetic resonance force microscopy (MRFM) is simulated, and the data statistics are shown to be those of a random telegraph signal with additive white noise. Larger-scale spin-dust models are simulated, having no spatial symmetry and no spatial ordering; the high-fidelity projection of numerically computed quantum trajectories onto low-dimensionality Kahler state-space manifolds is demonstrated. The reconstruction of quantum trajectories from sparse random projections is demonstrated, the onset of Donoho-Stodden breakdown at the Candes-Tao sparsity limit is observed, a deterministic construction for sampling matrices is given, and methods for quantum state optimization by Dantzig selection are given.Comment: 104 pages, 13 figures, 2 table

    Neuronal Assembly Detection and Cell Membership Specification by Principal Component Analysis

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    In 1949, Donald Hebb postulated that assemblies of synchronously activated neurons are the elementary units of information processing in the brain. Despite being one of the most influential theories in neuroscience, Hebb's cell assembly hypothesis only started to become testable in the past two decades due to technological advances. However, while the technology for the simultaneous recording of large neuronal populations undergoes fast development, there is still a paucity of analytical methods that can properly detect and track the activity of cell assemblies. Here we describe a principal component-based method that is able to (1) identify all cell assemblies present in the neuronal population investigated, (2) determine the number of neurons involved in ensemble activity, (3) specify the precise identity of the neurons pertaining to each cell assembly, and (4) unravel the time course of the individual activity of multiple assemblies. Application of the method to multielectrode recordings of awake and behaving rats revealed that assemblies detected in the cerebral cortex and hippocampus typically contain overlapping neurons. The results indicate that the PCA method presented here is able to properly detect, track and specify neuronal assemblies, irrespective of overlapping membership
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