A quantum de Finetti theorem in phase space representation

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).

5 pages

References in corpus (3)

• One-and-a-half quantum de Finetti theorems
• A Finite de Finetti Theorem for Infinite-Dimensional Systems
• On Exchangeable Continuous Variable Systems

Cited by in corpus (9)

• Gaussian Quantum Information
• Security of continuous-variable quantum key distribution against general attacks
• Continuous-variable quantum key distribution protocols with a non-Gaussian modulation
• Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations
• A simple proof that Gaussian attacks are optimal among collective attacks against continuous-variable quantum key distribution with a Gaussian modulation
• Mean-field limit of Bose systems: rigorous results
• De Finetti theorem on the CAR algebra
• Tight concentration inequalities for quantum adversarial setups exploiting permutation symmetry
• Gaussian work extraction from random Gaussian states is nearly impossible