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SU(p,q) coherent states and Gaussian de Finetti theorems

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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×n upper-left submatrix of an n×n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3=O(m).
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Dates and versions

hal-01656414 , version 1 (11-12-2017)

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  • HAL Id : hal-01656414 , version 1

Cite

Anthony Leverrier. SU(p,q) coherent states and Gaussian de Finetti theorems. QIP 2017 - 20th Annual Conference on Quantum Information Processing, Jan 2017, Seattle, United States. pp.1-24. ⟨hal-01656414⟩

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