24 research outputs found
Free probability of type B: analytic interpretation and applications
In this paper we give an analytic interpretation of free convolution of type
B, introduced by Biane, Goodman and Nica, and provide a new formula for its
computation. This formula allows us to show that free additive convolution of
type B is essentially a re-casting of conditionally free convolution. We put in
evidence several aspects of this operation, the most significant being its
apparition as an 'intertwiner' between derivation and free convolution of type
A. We also show connections between several limit theorems in type A and type B
free probability. Moreover, we show that the analytical picture fits very well
with the idea of considering type B random variables as infinitesimal
deformations to ordinary non-commutative random variables.Comment: 28 page
Random matrix techniques in quantum information theory
The purpose of this review article is to present some of the latest
developments using random techniques, and in particular, random matrix
techniques in quantum information theory. Our review is a blend of a rather
exhaustive review, combined with more detailed examples -- coming from research
projects in which the authors were involved. We focus on two main topics,
random quantum states and random quantum channels. We present results related
to entropic quantities, entanglement of typical states, entanglement
thresholds, the output set of quantum channels, and violations of the minimum
output entropy of random channels
Complete diagrammatics of the single ring theorem
Using diagrammatic techniques, we provide explicit functional relations
between the cumulant generating functions for the biunitarily invariant
ensembles in the limit of large size of matrices. The formalism allows to map
two distinct areas of free random variables: Hermitian positive definite
operators and non-normal R-diagonal operators. We also rederive the
Haagerup-Larsen theorem and show how its recent extension to the eigenvector
correlation function appears naturally within this approach.Comment: 18 pages, 6 figures, version accepted for publicatio
Spectrum of non-Hermitian heavy tailed random matrices
Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}|
is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our
main result is a heavy tailed counterpart of Girko's circular law. Namely,
under some additional smoothness assumptions on the law of X_{jk}, we prove
that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability
measure mu_alpha on C depending only on alpha such that with probability one,
the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1}
(X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our
approach combines Aldous & Steele's objective method with Girko's Hermitization
using logarithmic potentials. The underlying limiting object is defined on a
bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive
relations on the tree provide some properties of mu_alpha. In contrast with the
Hermitian case, we find that mu_alpha is not heavy tailed.Comment: Expanded version of a paper published in Communications in
Mathematical Physics 307, 513-560 (2011
Complex analysis methods in noncommutative probability
In this thesis we study convolutions that arise from noncommutative probability theory. We prove several regularity results for free convolutions, and for measures in partially defined one-parameter free convolution semigroups. We discuss connections between Boolean and free convolutions and, in the last chapter, we prove that any infinitely divisible probability measure with respect to monotonic additive or multiplicative convolution belongs to a one-parameter semigroup with respect to the corresponding convolution. Earlier versions of some of the results in this thesis have already been published, while some others have been submitted for publication. We have preserved almost entirely the specific format for PhD theses required by Indiana University. This adds several unnecessary pages to the document, but we wanted to preserve the specificity of the document as a PhD thesis at Indiana University
Free Bessel Laws
International audienceWe introduce and study a remarkable family of real probability measures that we call free Bessel laws. These are related to the free Poisson law. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups
Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
International audienceWe extend the so-called âsingle ring theoremâ (Feinberg and Zee 1997 Nucl. Phys. B 504 579), also known as the HaagerupâLarsen theorem (Haagerup and Larsen 2000 J. Funct. Anal. 176 331). We do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number. We give examples and provide a cross-check of the analytic prediction by the large scale numerics