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Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and…
In this paper, we develop the general formalism and properties of the spacetime density matrix, which captures correlations among different Cauchy surfaces and can be regarded as a natural generalization of the standard density matrix…
One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements, since it requires a global reconstruction. Here we experimentally demonstrate a scheme that can be used to…
Given a large real symmetric, positive semidefinite m-by-m matrix, the goal of this paper is to show how a numerical approximation of the entropy, given by the sum of the entropies of the individual eigenvalues, can be computed in an…
The Schmidt number represents the genuine entanglement dimension of a bipartite quantum state. We derive simple criteria for the Schmidt number of a density matrix in arbitrary local dimensions. They are based on the trace norm of…
Non-local properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values…
A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance. These pencils arise when…
Frozen Density Embedding Theory (FDET) [Wesolowski {\it Phys. Rev. A} {\bf 77}, 012504 (2008)] provides the interpretation of the eigenvalue equations for an embedded $N'$-electron wavefunction, in which the embedding operator is…
We consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes. The physical phase space of such spacetimes is spanned by the mass and the charge parameters $M$ and $Q$ of the…
We extend balloon and sample-smoothing estimators, two types of variable-bandwidth kernel density estimators, by a shift parameter and derive their asymptotic properties. Our approach facilitates the unified study of a wide range of density…
This paper investigates the approximation of invariant measures for McKean-Vlasov stochastic differential equations (SDEs) using the Euler-Maruyama (EM) scheme under a monotonicity condition. Firstly, the convergence of the numerical…
A new method of solution to the local spin density approximation to the electronic Schr\"{o}dinger equation is presented. The method is based on an efficient, parallel, adaptive multigrid eigenvalue solver. It is shown that adaptivity is…
With the emergence of Artificial Intelligence, numerical algorithms are moving towards more approximate approaches. For methods such as PCA or diffusion maps, it is necessary to compute eigenvalues of a large matrix, which may also be dense…
The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the R\'enyi entropy of index $m$, which captures the higher moments of the reduced density matrix. In this work, we…
As far as entanglement is concerned, two density matrices of $n$ particles are equivalent if they are on the same orbit of the group of local unitary transformations, $U(d_1)\times...\times U(d_n)$ (where the Hilbert space of particle $r$…
The Lewenstein-Sanpera decomposition for a generic two-qubit density matrix is obtained by using Wootters's basis. It is shown that the average concurrence of the decomposition is equal to the concurrence of the state. It is also shown that…
We investigate random density matrices obtained by partial tracing larger random pure states. We show that there is a strong connection between these random density matrices and the Wishart ensemble of random matrix theory. We provide…
Although quantum entanglement has already been verified experimentally and applied in quantum computing, quantum sensing and quantum networks, most of the existing measures cannot characterize the entanglement faithfully. In this work, by…
We develop techniques to compute the k-th Moment of the Eigenvalue-statistic for a random Matrix M the entries of which do not have to be necessarily Independent. The dependence is controlled via an equivalence relation on the pairs of the…
The Nystr\"{o}m method is routinely used for out-of-sample extension of kernel matrices. We describe how this method can be applied to find the singular value decomposition (SVD) of general matrices and the eigenvalue decomposition (EVD) of…