Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

arXiv:1601.02365Reviews in Mathematical PhysicsMar 20, 201712 citations

On quasiequivalence of locally normal states

Александр Семенович Холево

Theoretical and Mathematical PhysicsNov 1, 197262 citations

Finite de Finetti theorem for conditional probability distributions describing physical theories

Matthias Christandl, Ben Toner

We work in a general framework where the state of a physical system is defined by its behavior under measurement and the global state is constrained by no-signaling conditions. We show that the marginals of symmetric states in such theories can be approximated by convex combinations of independent and identical conditional probability distributions, generalizing the classical finite de Finetti theorem of Diaconis and Freedman. Our results apply to correlations obtained from quantum states even when there is no bound on the local dimension, so that known quantum de Finetti theorems cannot be used.We work in a general framework where the state of a physical system is defined by its behavior under measurement and the global state is constrained by no-signaling conditions. We show that the marginals of symmetric states in such theories can be approximated by convex combinations of independent and identical conditional probability distributions, generalizing the classical finite de Finetti theorem of Diaconis and Freedman. Our results apply to correlations obtained from quantum states even when there is no bound on the local dimension, so that known quantum de Finetti theorems cannot be used.

arXiv:0712.0916Journal of Mathematical PhysicsApr 1, 200924 citations

$SU(p,q)$ coherent states and a Gaussian de Finetti theorem$SU(p,q)$ coherent states and a Gaussian de Finetti theorem

Anthony Leverrier

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 $n\times n$ upper-left submatrix of an $n\times n$ Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if $n^3 =O(m)$ .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 $n\times n$ upper-left submatrix of an $n\times n$ Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if $n^3 =O(m)$.

arXiv:1612.05080Dec 15, 20162 citations

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

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\times n

upper-left submatrix of an

n\times n

Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if

n^3 =O(m)