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arXiv:0904.4862 · Physical Review A

Quantum de Finetti theorem in phase-space representation

Anthony Leverrier, Nicolas J. Cerf

Published Jul 28, 2009• 11 citations
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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, toward probabilistic mixtures of independent and identically distributed (IID) states of the form ${\ensuremath{\sigma}}^{\ensuremath{\bigotimes}n}$. 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).

DOI
10.1103/physreva.80.010102

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The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires the original state to obey permutation symmetry conditioned on successful experimental verification on k of N subsystems. We generalize the de Finetti theorem to include asymmetric bounds on the variance of canonical observables and biased basis selection during the verification step. Our result thereby enables application of infinite-dimensional de Finetti theorem to situations where two conjugate measurements obey different statistics, such as the security analysis of quantum key distribution protocols based on squeezed state against coherent attack.

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The quantum versions of de Finetti's theorem derived so far express the convergence of nnn-partite symmetric states, i.e., states that are invariant under permutations of their nnn parties, toward probabilistic mixtures of independent and identically distributed (IID) states of the form \ensuremathσ\ensuremath⨂n{\ensuremath{\sigma}}^{\ensuremath{\bigotimes}n}\ensuremathσ\ensuremath⨂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 nnn-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, nnn identical thermal states).
SU(p,q)SU(p,q)SU(p,q) coherent states and a Gaussian de Finetti theorem
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 nnn copies of that space, we consider the action of the unitary group U(n)U(n)U(n) on the creation operators of the nnn modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in U(n)U(n)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)SU(p,q)SU(p,q) of signature (p,q)(p,q)(p,q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p,q)SU(p,q)SU(p,q). We therefore find a dual unitary representation of the pair of groups U(n)U(n)U(n) and SU(p,q)SU(p,q)SU(p,q) on an n(p+q)n(p+q)n(p+q)-mode Fock space. The (Gaussian) SU(p,q)SU(p,q)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×nn\times nn×n upper-left submatrix of an n×nn\times nn×n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3=O(m)n^3 =O(m)n3=O(m).
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).
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).
Standard finite quantum de Finetti theorems approximate the kkk-system marginals of permutation-invariant states of nnn-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our k/nk/nk/n error bound in max-relative entropy improves on the previously best known k2/nk^2/nk2/n scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels N(n):A⊗n→B⊗nN^{(n)}:A^{\otimes n}\to B^{\otimes n}N(n):A⊗n→B⊗n, where dA=dim⁡Ad_A=\dim AdA​=dimA, we prove that the kkk-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n)O(k/\sqrt{n})O(k/n​) and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of rrr defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2\mathrm{poly}(n)\bigl(2d_A^4k^3/(nr^2)\bigr)^{(r+1)/2}poly(n)(2dA4​k3/(nr2))(r+1)/2 and decays exponentially in nnn for a suitable choice of parameters rrr and kkk.
According to the Quantum de Finetti Theorem, locally normal infinite particle states with Bose-Einstein symmetry can be represented as mixtures of infinite tensor powers of vector states. This note presents examples of infinite-particle states with Bose-Einstein symmetry that arise as limits of Gibbs ensembles on finite dimensional spaces, and displays their de Finetti representations. We consider Gibbs ensembles for systems of bosons in a finite dimensional setting and discover limits as the number of particles tends to infinity, provided the temperature is scaled in proportion to particle number.
The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires the original state to obey permutation symmetry conditioned on successful experimental verification on k of N subsystems. We generalize the de Finetti theorem to include asymmetric bounds on the variance of canonical observables and biased basis selection during the verification step. Our result thereby enables application of infinite-dimensional de Finetti theorem to situations where two conjugate measurements obey different statistics, such as the security analysis of quantum key distribution protocols based on squeezed state against coherent attack.