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

TL;DR

This paper generalizes the quantum de Finetti theorem to infinite-dimensional Fock spaces using $SU(p,q)$ coherent states, revealing new unitary representations and implications for random matrix theory.

Contribution

It introduces a new class of coherent states related to $SU(p,q)$, characterizes the invariant subspace under $U(n)$, and proves a Gaussian de Finetti theorem for unitary-invariant states.

Findings

Characterization of the invariant subspace as $SU(p,q)$ coherent states.

Proof of a Gaussian de Finetti theorem for unitary-invariant states.

Approximation of Haar-invariant matrices by independent normal variables.

Abstract

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 $n^3 =O(m)$.