Local Eigenvalue Density for General MANOVA Matrices

We consider random n×n matrices of the form $$\begin{aligned} \left( XX^*+YY^*\right)^{-\frac{1}{2}}YY^*\left( XX^*+YY^*\right )^{-\frac{1}{2}} , \end{aligned}$$ where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to logn factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices.     

Quantum Entanglement and Reduced Density Matrices

We investigate entanglement and separability criteria of multipartite (n-partite) state by examining ranks of its reduced density matrices. Firstly, we construct the general formula to determine the criterion. A rank of origin density matrix always equals one, meanwhile ranks of reduced matrices have various ranks. Next, separability and entanglement criterion of multipartite is determined by calculating ranks of reduced density matrices. In this article we diversify multipartite state criteria into completely entangled state, completely separable state, and compound state, i.e. sub-entangled state and sub-entangledseparable state. Furthermore, we also shorten the calculation proposed by the previous research to determine separability of multipartite state and expand the methods to be able to differ multipartite state based on criteria above.     

A Note on the Eigenvalue Density of Random Matrices

The distribution of eigenvalues of N 2 N random matrices in the limit N M X is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a consequence of a more general theorem, proven here, in the statistical mechanics of unstable interactions. Our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems.     

Entropy and Entanglement Bounds for Reduced Density Matrices of Fermionic States

Unlike bosons, fermions always have a non-trivial entanglement. Intuitively, Slater determinantal states should be the least entangled states. To make this intuition precise we investigate entropy and entanglement of fermionic states and prove some extremal and near extremal properties of reduced density matrices of Slater determinantal states.     

On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices

We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent.     

Gas Density Distributions and Reduced Field Strenghts of the Positive Column of Low Pressure XeCl* Glow Discharges

Interferometric measurements of radial gas density distributions have been performed on the cylindrical positive column of DC low pressure glow discharges (LPGD) in pure Xe and Xe/Cl₂ gas mixtures. Absolute gas temperatures have been measured by thermocouples. In the mixtures, the gas temperature is several hundred Kelvin above the temperature in pure Xe. Additionally, the radial distribution of the gas density in the mixtures cannot be described by Bessel profiles, which would result from Schottky's diffusion theory. Combined with field strength measurements, radial profiles of E/N (electric field strength/neutral density) have been determined. Results of this work will be useful for model developments of LPGD in rare-gas/Cl₂ mixtures but also for the general understanding of the positive column in attaching gases.     

The Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices

The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue for some non-necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov (cf. [11]) in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.     

Multipartite Separability of Density Matrices of Graphs

A new layers method is presented for multipartite separability of density matrices from simple graphs. Full separability of tripartite states is studied for graphs on degree symmetric premise. The models are generalized to multipartite systems by presenting a class of fully separable states arising from partially symmetric graphs.     

Density of Eigenvalues of Random Normal Matrices

The relation between random normal matrices and conformal mappings discovered by Wiegmann and Zabrodin is made rigorous by restricting normal matrices to have spectrum in a bounded set. It is shown that for a suitable class of potentials the asymptotic density of eigenvalues is uniform with support in the interior domain of a simple smooth curve.     

Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices

In this paper, we study the complex Wigner matrices $M_{n}=\frac{1}{\sqrt{n}}W_{n}$ whose eigenvalues are typically in the interval [?2,2]. Let λ ₁≤λ ₂?≤λ _n be the ordered eigenvalues of M _n . Under the assumption of four matching moments with the Gaussian Unitary Ensemble (GUE), for test function f 4-times continuously differentiable on an open interval including [?2,2], we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold u in the bulk of the Wigner semicircle law as $\mathcal{A}_{n}[f; u]=\sum_{l=1}^{n}f(\lambda_{l})\mathbf{1}_{\{\lambda_{l}\leq u\}}$ . And the second one is $\mathcal{B}_{n}[f; k]=\sum_{l=1}^{k}f(\lambda_{l})$ with positive integer k=k _n such that k/n→y∈(0,1) as n tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from $\mathcal{B}_{n}[f; \lfloor nt\rfloor]$ . The main difficulty is to deal with the linear eigenvalue statistics for the test functions with several non-differentiable points. And our main strategy is to combine the Helffer-Sjöstrand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on $\mathcal{A}_{n}[f;u]$ for the real Wigner matrices will also be briefly discussed.     

Density of States for Random Band Matrices

 By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W ² , for W large but fixed. Received: 6 February 2002 / Accepted: 17 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by NSF grant DMS 9729992     

Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge

This is a continuation of our earlier paper (Tao and Vu, , 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (, 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697–733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.     

Transfer Matrices and Lattice Fermions at Finite Density

I discuss the connection between the Hamiltonian and path integral approaches for fermionic fields. I show how the temporal Wilson projection operators appear naturally in a lattice action. I also carefully treat the insertion of a chemical potential term.     

New Inequality for Density Matrices of Single Qudit States

Using the monotonicity of relative entropy of composite quantum systems, we obtain new entropic inequalities for arbitrary density matrices of single qudit states. Examples of qutrit state inequalities and the “qubit portrait” bound for the distance between the qutrit states are considered in explicit form.     

Random Incidence Matrices: Moments of the Spectral Density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of small eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.     

On the Role of Density Matrices in Bohmian Mechanics

It is well known that density matrices can be used in quantum mechanics to represent the information available to an observer about either a system with a random wave function (statistical mixture) or a system that is entangled with another system (reduced density matrix). We point out another role, previously unnoticed in the literature, that a density matrix can play: it can be the conditional density matrix, conditional on the configuration of the environment. A precise definition can be given in the context of Bohmian mechanics, whereas orthodox quantum mechanics is too vague to allow a sharp definition, except perhaps in special cases. In contrast to statistical and reduced density matrices, forming the conditional density matrix involves no averaging. In Bohmian mechanics with spin, the conditional density matrix replaces the notion of conditional wave function, as the object with the same dynamical significance as the wave function of a Bohmian system.PACS number:03.65.Ta (foundations of quantum mechanics)     

Gaussian Density Matrices: Quantum Analogs of Classical States

We study quantum analogs of classical situations, i.e. quantum states possessing some specific classical attribute(s). These states seem quite generally, to have the form of gaussian density matrices. Such states can always be parametrized as thermal squeezed states (TSS). We consider the following specific cases: (a) Two beams that are built from initial beams which passed through a beam splitter cannot, classically, be distinguished from (appropriately prepared) two independent beams that did not go through a splitter. The only quantum states possessing this classical attribute are TSS. (b) The classical Cramer's theorem was shown to have a quantum version (Hegerfeldt). Again, the states here are Gaussian density matrices. (c) The special case in the study of the quantum version of Cramer's theorem, viz. when the state obtained after partial tracing is a pure state, leads to the conclusion that all states involved are zero temperature limit TSS. The classical analog here are gaussians of zero width, i.e. all distributions are δ functions in phase space.