A quantum de Finetti theorem in phase space representation

TL;DR

This paper introduces a new quantum de Finetti theorem for continuous-variable systems, showing that phase space symmetry invariance leads to convergence towards mixtures of i.i.d. Gaussian states, extending the theorem's applicability.

Contribution

It proposes a novel form of quantum de Finetti theorem based on phase space invariance, applicable to infinite-dimensional systems unlike previous permutation-based versions.

Findings

Invariance under phase space orthogonal transformations implies convergence to i.i.d. Gaussian states.

The theorem applies to n-mode bosonic states with phase space symmetry.

It extends de Finetti's theorem to infinite-dimensional, continuous-variable quantum systems.

Abstract

The quantum versions of de Finetti's theorem derived so far express the convergence of n-partite symmetric states, i.e., states that are invariant under permutations of their n parties, towards probabilistic mixtures of independent and identically distributed (i.i.d.) states. Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads to a new type of quantum de Finetti's theorem that is particularly relevant to continuous-variable systems. Specifically, an n-mode bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge towards a probabilistic mixture of i.i.d. Gaussian states (actually, n identical thermal states).