Let S⊂Zd be a finitely generated subsemigroup. Let E be a product system over S. We show that there exists an infinite dimensional separable Hilbert space H and a semigroup α:={αx}x∈S of unital ...normal ∗-endomorphisms of B(H) such that E is isomorphic to the product system associated to α.
Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions and applications of distance ...correlation have been discussed in the recent literature, but the problem of defining the partial distance correlation has remained an open question of considerable interest. The problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space. For the definition of partial distance correlation, we introduce a new Hubert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hubert space, and develop and implement a test for zero partial distance correlation. Our intermediate results provide an unbiased estimator of squared distance covariance, and a neat solution to the problem of distance correlation for dissimilarities rather than distances.
Modeling of uncertainty differential equations is very important issue in applied sciences and engineering, while the natural way to model such dynamical systems is to use fuzzy differential ...equations. In this paper, we present a new method for solving fuzzy differential equations based on the reproducing kernel theory under strongly generalized differentiability. The analytic and approximate solutions are given with series form in terms of their parametric form in the space
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The method used in this paper has several advantages; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, needs less effort to achieve the results, and is developed especially for the nonlinear cases; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions and their derivatives will be applicable; fourth, the method does not require discretization of the variables, and it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. Results presented in this paper show potentiality, generality, and superiority of our method as compared with other well-known methods.
Dual-unitary quantum circuits can be used to construct 1+1 dimensional lattice models for which dynamical correlations of local observables can be explicitly calculated. We show how to analytically ...construct classes of dual-unitary circuits with any desired level of (non-)ergodicity for any dimension of the local Hilbert space, and present analytical results for thermalization to an infinite-temperature Gibbs state (ergodic) and a generalized Gibbs ensemble (nonergodic). It is shown how a tunable ergodicity-inducing perturbation can be added to a nonergodic circuit without breaking dual unitarity, leading to the appearance of prethermalization plateaux for local observables.
A dynamical quantum simulation of SU(2) non-Abelian gauge field theory on a digital quantum computer is presented. This was enabled on current quantum hardware by introducing a mapping of the field ...onto a register of qubits that utilizes local gauge symmetry while preserving local constraints on the fields, reducing the dimensionality of the calculation. Controlled plaquette operators and gauge-variant completions in the unphysical part of the Hilbert space were designed and used to implement time evolution. The new techniques developed in this work generalize to quantum simulations of higher dimensional gauge field theories.
In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the ...eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e., Hilbert space of size N≃2.7×106). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.
Abstract
The goal of this paper, is to introduce another classes of the fuzzy soft bounded linear operator in the fuzzy soft Hilbert space which is a fuzzy soft quasi normal operator, as well as, ...give some properties about this concept with investigating the relationship among this types of the fuzzy soft bounded linear operator on fuzzy soft Hilbert space with other kinds of fuzzy soft bounded linear operators.
In experimentally realistic situations, quantum systems are never perfectly isolated and the coupling to their environment needs to be taken into account. Often, the effect of the environment can be ...well approximated by a Markovian master equation. However, solving this master equation for quantum many-body systems becomes exceedingly hard due to the high dimension of the Hilbert space. Here we present an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine-learning techniques. We represent the mixed many-body quantum states with neural networks in the form of restricted Boltzmann machines and derive a variational Monte Carlo algorithm for their time evolution and stationary states. We document the accuracy of the approach with numerical examples for a dissipative spin lattice system.
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