Joint Extension of States of Subsystems for a CAR System

Abstract

The problem of existence and uniqueness of a state of a joint system with given restrictions to subsystems is studied for a Fermion system, where a novel feature is non-commutativity between algebras of subsystems.For an arbitrary (finite or infinite) number of given subsystems, a product state extension is shown to exist if and only if all states of subsystems except at most one are even (with respect to the Fermion number). If the states of all subsystems are pure, then the same condition is shown to be necessary and sufficient for the existence of any joint extension. If the condition holds, the unique product state extension is the only joint extension.For a pair of subsystems, with one of the given subsystem states pure, a necessary and sufficient condition for the existence of a joint extension and the form of all joint extensions (unique for almost all cases) are given. For a pair of subsystems with non-pure subsystem states, some classes of examples of joint extensions are given where non-uniqueness of joint extensions prevails.

De Finetti Theorem on the CAR Algebra

“…Fix ϕ 1 ∈ S(CAR(I 1 )), ϕ 2 ∈ S(CAR(I 2 )). If at least one among them is even, then according to Theorem 1 of [5], the product state extension (called product state for short) ϕ ∈ S(CAR(I 1 I 2 )) is uniquely defined. We write, with an abuse of notation, ϕ = ϕ 1 ϕ 2 .…”

“…By Theorem 1 of [5], any product state is uniquely determined by the product of the values of the state on the generators. Hence, it is enough to check, for each n ∈ N and A 1 , .…”

“…The introduction and the study of the notion of the product state, and the application to general thermodynamical properties of Fermi lattice systems (see e.g. [4,5] and the references cited therein). The connection with the Markov Property, Quantum Statistical Systems, Quantum Information Theory and Entanglement, of chains of Fermi systems [2,16].…”

On separable states for composite systems of distinguishable fermions

“…This implies that ω i has pure state restrictions on both A(I) and A(J). By Theorem 1 (2) in [7], at least one of ω i | A(I) and ω i | A(J) should be even for the existence of their state extension ω i on A(I ∪ J) and ω i is uniquely given as…”

“…It is, however, important to note that due to the CAR structure (algebraic nonindependence) there are limitations on marginal states that can be prepared on disjoint regions [7] and hence on product states.…”