24,582 research outputs found
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
On the volume of the set of mixed entangled states
A natural measure in the space of density matrices describing N-dimensional
quantum systems is proposed. We study the probability P that a quantum state
chosen randomly with respect to the natural measure is not entangled (is
separable). We find analytical lower and upper bounds for this quantity.
Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate
that P decreases exponentially with N. Analysis of a conditional measure of
separability under the condition of fixed purity shows a clear dualism between
purity and separability: entanglement is typical for pure states, while
separability is connected with quantum mixtures. In particular, states of
sufficiently low purity are necessarily separable.Comment: 10 pages in LaTex - RevTex + 4 figures in eps. submitted to Phys.
Rev.
Eigenvalue Distributions of Reduced Density Matrices
Given a random quantum state of multiple distinguishable or indistinguishable
particles, we provide an effective method, rooted in symplectic geometry, to
compute the joint probability distribution of the eigenvalues of its one-body
reduced density matrices. As a corollary, by taking the distribution's support,
which is a convex moment polytope, we recover a complete solution to the
one-body quantum marginal problem. We obtain the probability distribution by
reducing to the corresponding distribution of diagonal entries (i.e., to the
quantitative version of a classical marginal problem), which is then determined
algorithmically. This reduction applies more generally to symplectic geometry,
relating invariant measures for the coadjoint action of a compact Lie group to
their projections onto a Cartan subalgebra, and can also be quantized to
provide an efficient algorithm for computing bounded height Kronecker and
plethysm coefficients.Comment: 51 pages, 7 figure
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