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arXiv · 0904.4862

A quantum de Finetti theorem in phase space representation

Anthony Leverrier·Nicolas J. Cerf
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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).

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Anthony Leverrier, Nicolas J. Cerf. 2009-05-04. A quantum de Finetti theorem in phase space representation. https://doi.org/10.1103/physreva.80.010102

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