Quantum de Finetti theorem in phase-space representation

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, toward probabilistic mixtures of independent and identically distributed (IID) states of the form ${\sigma }^{\otimes n}$. 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 quantum de Finetti theorem 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 toward a probabilistic mixture of IID Gaussian states (actually, $n$ identical thermal states).