De Finetti theorem on the CAR algebra

Abstract

The symmetric states on a quasi local C*-algebra on the infinite set of indices J are those invariant under the action of the group of the permutations moving only a finite, but arbitrary, number of elements of J. The celebrated De Finetti Theorem describes the structure of the symmetric states (i.e. exchangeable probability measures) in classical probability. In the present paper we extend De Finetti Theorem to the case of the CAR algebra, that is for physical systems describing Fermions. Namely, after showing that a symmetric state is automatically even under the natural action of the parity automorphism, we prove that the compact convex set of such states is a Choquet simplex, whose extremal (i.e. ergodic w.r.t. the action of the group of permutations previously described) are precisely the product states in the sense of Araki-Moriya. In order to do that, we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self--containing interest.

23 pages, juornal reference: Communications in Mathematical Physics, to appear

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Cited by in corpus (13)

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• Ground States of Fermionic lattice Hamiltonians with Permutation Symmetry
• Wick order, spreadability and exchangeability for monotone commutation relations
• A note on Boolean stochastic processes
• $C^{\ast }$-fermi systems and detailed balance
• Ergodic properties of the Anzai skew-product for the noncommutative torus
• A fermionic de Finetti theorem
• Symmetric states for $C^{\ast }$-Fermi systems I: De Finetti theorem
• Symmetric states for $C^{\ast }$-Fermi systems II: Klein transformation and their structure
• A noncommutative De Finetti theorem for boolean independence
• Fermionic quantum detailed balance and entanglement
• De Finetti Theorems for Braided Parafermions
• Exchangeable stochastic processes and symmetric states in quantum probability