Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.09494

Infinite twisted $C^*$-tensor product and symmetric states

Francesco Fidaleo, Elia Vincenzi

Published Sep 8, 2026• 0 citations
arXivHTMLPDF

Abstract

The twisted $C^*$-tensor product was exhaustively investigated by the authors in two previous papers. In this new context, generalising the usual tensor product and the Fermi one, we analyse the possibility of studying the set of symmetric states, that is those invariant under all finite permutations, in the setting of infinite twisted $C^*$-tensor products. As a preliminary result, we recognise that such an investigation can proceed only when the bicharacter, involved in the construction of such twisted tensor products, is hermitian. Otherwise, there is no natural action of the finitary symmetric group on the infinite twisted chain. Furthermore, even if the investigation of symmetric states is certainly meaningful for all twisted systems based on hermitian bicharacters, it is shown that it can be fruitfully carried out in three cases only. In these cases, we can provide the "genuine" version of the celebrated De Finetti Theorem already established by Hewitt and Savage for the general classical case, Stormer for the usual tensor product and Fidaleo for Fermi models. Very surprisingly, it emerges that one more twisted model can be treated exhaustively: it corresponds to the chain twisted by the so-called Klein four-group, and its Klein bicharacter unique up to equivalence.

DOI
10.48550/arxiv.2609.09494

Related papers

De Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products

Vitonofrio Crismale, Stefano Rossi, Paola Zurlo

Local actions of [Formula: see text], the group of finite permutations on [Formula: see text], on quasi-local algebras are defined and proved to be [Formula: see text]-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of [Formula: see text]-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon [Formula: see text] in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here.

arXiv:2201.02488•Infinite Dimensional Analysis Quantum Probability and Related Topics•Nov 18, 2022•5 citations

Symmetric states for $C^*$-Fermi systems I: De Finetti theorem

Francesco Fidaleo

n the present note, which is the first part of a work concerning the study of the set of the symmetric states for Fermi systems, we describe the extension of the De Finetti theorem to the infinite Fermi $C^*$-tensor product of a single (separable) general $\bz^2$-graded $C^*$-algebra.

arXiv:2201.00421•Jan 2, 2022•1 citation

On Fermi Quantum Markov Chains

Francesco Fidaleo

Abstract We deal with the analogous of the so-called Quantum Markov Chains for Fermi systems (i.e. the models whose fundamental fields enjoy the Canonical Anti-commutation Relations), and discuss their main properties. Since the involution and the product highly differ from those of the usual tensor product, the positivity of the involved operators and functionals, not automatically satisfied, is a very delicate question. As a consequence, the positivity should be checked directly for the specific models under consideration. Therefore, we analyse some pivotal cases, providing examples which go beyond the simpler ones already studied in literature. As an interesting result, we prove that the symmetric states, that is those which are invariant under the natural action of the group of all finite permutations, are described in terms of particular Quantum Markov Chains.

International Journal of Theoretical Physics•Dec 4, 2025•0 citations

$SU(p,q)$ 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 $n^3 =O(m)$.

arXiv:1612.05080•Dec 15, 2016•2 citations

A most compendious and facile quantum de Finetti theorem

Robert König, Graeme Mitchison

In its most basic form, the finite quantum de Finetti theorem states that the reduced k-partite density operator of an n-partite symmetric state can be approximated by a convex combination of k-fold product states. Variations of this result include Renner’s “exponential” approximation by “almost-product” states, a theorem which deals with certain triples of representations of the unitary group, and the result of D’Cruz et al. [e-print quant-ph/0606139;Phys. Rev. Lett. 98, 160406 (2007)] for infinite-dimensional systems. We show how these theorems follow from a single, general de Finetti theorem for representations of symmetry groups, each instance corresponding to a particular choice of symmetry group and representation of that group. This gives some insight into the nature of the set of approximating states and leads to some new results, including an exponential theorem for infinite-dimensional systems.

arXiv:quant-ph/0703210•Journal of Mathematical Physics•Jan 1, 2009•29 citations

A de Finetti representation for finite symmetric quantum states

Robert König, Renato Renner

Consider a symmetric quantum state on an n-fold product space, that is, the state is invariant under permutations of the n subsystems. We show that, conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems, the state in the remaining subsystems is close to having product form. This immediately generalizes the so-called de Finetti representation to the case of finite symmetric quantum states.

arXiv:quant-ph/0410229•Journal of Mathematical Physics•Dec 1, 2005•124 citations
Infinite twisted C∗C^*C∗-tensor product and symmetric states
The twisted C∗C^*C∗-tensor product was exhaustively investigated by the authors in two previous papers. In this new context, generalising the usual tensor product and the Fermi one, we analyse the possibility of studying the set of symmetric states, that is those invariant under all finite permutations, in the setting of infinite twisted C∗C^*C∗-tensor products. As a preliminary result, we recognise that such an investigation can proceed only when the bicharacter, involved in the construction of such twisted tensor products, is hermitian. Otherwise, there is no natural action of the finitary symmetric group on the infinite twisted chain. Furthermore, even if the investigation of symmetric states is certainly meaningful for all twisted systems based on hermitian bicharacters, it is shown that it can be fruitfully carried out in three cases only. In these cases, we can provide the "genuine" version of the celebrated De Finetti Theorem already established by Hewitt and Savage for the general classical case, Stormer for the usual tensor product and Fidaleo for Fermi models. Very surprisingly, it emerges that one more twisted model can be treated exhaustively: it corresponds to the chain twisted by the so-called Klein four-group, and its Klein bicharacter unique up to equivalence.
Local actions of [Formula: see text], the group of finite permutations on [Formula: see text], on quasi-local algebras are defined and proved to be [Formula: see text]-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of [Formula: see text]-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon [Formula: see text] in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here.
Symmetric states for C∗C^*C∗-Fermi systems I: De Finetti theorem
n the present note, which is the first part of a work concerning the study of the set of the symmetric states for Fermi systems, we describe the extension of the De Finetti theorem to the infinite Fermi C∗C^*C∗-tensor product of a single (separable) general $\bz^2$-graded C∗C^*C∗-algebra.
Abstract We deal with the analogous of the so-called Quantum Markov Chains for Fermi systems (i.e. the models whose fundamental fields enjoy the Canonical Anti-commutation Relations), and discuss their main properties. Since the involution and the product highly differ from those of the usual tensor product, the positivity of the involved operators and functionals, not automatically satisfied, is a very delicate question. As a consequence, the positivity should be checked directly for the specific models under consideration. Therefore, we analyse some pivotal cases, providing examples which go beyond the simpler ones already studied in literature. As an interesting result, we prove that the symmetric states, that is those which are invariant under the natural action of the group of all finite permutations, are described in terms of particular Quantum Markov Chains.
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).
In its most basic form, the finite quantum de Finetti theorem states that the reduced k-partite density operator of an n-partite symmetric state can be approximated by a convex combination of k-fold product states. Variations of this result include Renner’s “exponential” approximation by “almost-product” states, a theorem which deals with certain triples of representations of the unitary group, and the result of D’Cruz et al. [e-print quant-ph/0606139;Phys. Rev. Lett. 98, 160406 (2007)] for infinite-dimensional systems. We show how these theorems follow from a single, general de Finetti theorem for representations of symmetry groups, each instance corresponding to a particular choice of symmetry group and representation of that group. This gives some insight into the nature of the set of approximating states and leads to some new results, including an exponential theorem for infinite-dimensional systems.