SU(p,q)SU(p,q) coherent states and a Gaussian de Finetti theorem

We prove a generalization of the quantum de Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of considering the action of the permutation group on $n$ copies of that space, we consider the action of the unitary group $U(n)$ on the creation operators of the $n$ modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in $U(n)$. Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group $SU(p,q)$ of signature $(p,q)$. More precisely, this construction yields a unitary representation of the noncompact simple real Lie group $SU(p,q)$. We therefore find a dual unitary representation of the pair of groups $U(n)$ and $SU(p,q)$ on an $n(p+q)$-mode Fock space. The (Gaussian) $SU(p,q)$ coherent states resolve the identity on the symmetric subspace, which implies a Gaussian de Finetti theorem stating that tracing over a few modes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this de Finetti theorem, we show that the $n\times n$ upper-left submatrix of an $n\times n$ Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if $n^3 =O(m)$.Comment: v2: 39 pages, including new application to truncations of Haar random matrices. Comments are welcom

Deflating quadratic matrix polynomials with structure preserving transformations

AbstractGiven a pair of distinct eigenvalues (λ1,λ2) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form Qd(λ)00q(λ) having the same eigenvalue s as Q(λ), with Qd(λ) an (n-1)×(n-1) quadratic matrix polynomial and q(λ) a scalar quadratic polynomial with roots λ1 and λ2. This block diagonalization cannot be achieved by a similarity transformation applied directly to Q(λ) unless the eigenvectors corresponding to λ1 and λ2 are parallel. We identify conditions under which we can construct a family of 2n×2n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q(λ), (c) preserve the block structure of a large class of block symmetric linearizations of Q(λ), thereby defining new quadratic matrix polynomials Q1(λ) that have the same eigenvalue s as Q(λ), (d) yield quadratics Q1(λ) with the property that their eigenvectors associated with λ1 and λ2 are parallel and hence can subsequently be deflated by a similarity applied directly to Q1(λ). This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action

Asymptotic solutions of almost diagonal differential and difference systems

New methods for both asymptotic integration of the linear differential systems Y\u27(t) = [D( t) + R(t)]Y( t) and asymptotic summation of the linear difference systems Y(t + 1) = [D(t)+ R(t)]Y(t) are derived. The fundamental solution Y(t) = phi( t)[I+P(t)] for differential and difference systems is constructed in terms of a product. The first matrix function phi(t) is decided by the diagonal matrix D(t) and the second matrix I + P(t) is a perturbation of the identity matrix I. Another fundamental solution Y( t) = [I + Q(t)]phi( t) is also constructed for difference systems. Conditions are given on the matrix [D(t) + R( t)] that allow us to represent I + P( t) or Q(t) + I as an absolutely convergent resolvent series without imposing stringent conditions on R(t). In particular the analogs, in the setting of difference equations, of fundamental theorems of Levison and Hartman-Wintner are shown to follow from one and same theorem in this work