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Infinite Dimensional Analysis Quantum Probability and Related Topics

The classical de Finetti theorem for strongly quasi-invariant states

Luigi Accardi, Ameur Dhahri

Published Jul 28, 2026• 0 citations
Publisher

Abstract

As an application of a general theory of quasi-invariant states developed in Ref. [1], we prove a classical de Finetti type structure theorem for quasi-invariant states and, for this class of states, we also extend the conditional version of this theorem, first proved in Ref. [2].
DOI
10.1142/s0219025726500141

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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.
De Finetti theorems convert permutation symmetry into approximate mixtures of product states and thereby justify a wide range of reductions in classical and quantum statistics. In this work we study de Finetti hierarchies in which the feasible states are additionally constrained to be fixed points of quantum channels, a condition that subsumes invariance under arbitrary compact symmetry groups. Combining the mean-ergodic theorem with the structure theory of conditional expectations, we prove a tight bound on the entanglement-assisted classical capacity of the dual of a conditional expectation, block-wise distortion bounds for informationally complete measurements adapted to fixed point algebras, and an exact type-based refinement of the chain rule for permutation-invariant states. From these tools we derive several de Finetti theorems: a double-sided extension theorem with O(log⁡n/n)O\left(\sqrt{\log n}/n\right)O(logn​/n) convergence, an interpolation theorem whose dimension dependence is governed solely by the block structure of the fixed point algebras and which recovers the dimension-independent classical behavior for maximal tori, and a Bose-symmetric variant. Exploiting Schur-Weyl duality and Gelfand-Tsetlin bases, we further show that the rounding scheme producing certifiably good separable inner approximations for (constrained) separability problems can be implemented in time polynomial in 1/ε1/ε1/ε for fixed local dimensions, complementing the known efficient outer hierarchies. Applications to bilinear optimization under symmetries and to approximate quantum error correction are discussed.
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.
de Finetti-type theorems enable a substantially simplified analysis of information-processing tasks in various areas, such as quantum cryptography and quantum tomography. The idea is that instead of carrying out the analysis for any possible state it is sufficient to consider one particular de Finetti state, that is, a convex combination of i.i.d. states. It is thus interesting to see whether such de Finetti-type theorems are unique for quantum theory or whether they hold for more general theories. Here we prove a de Finetti-type theorem in the framework of conditional probability distributions. In this framework, a system is described by a conditional probability distribution P_A|X where X denotes the measurement and A the outcome. We show that any permutation invariant system P_A|X can be reduced to a de Finetti system. We then demonstrate how the theorem can be applied to security proofs of device independent cryptographic protocols.