Joint Extension of States of Subsystems for a CAR System

TL;DR

This paper investigates the conditions under which joint states of Fermion systems can be extended from subsystem states, revealing that non-commutativity and parity conditions determine existence and uniqueness of such extensions.

Contribution

It provides necessary and sufficient conditions for the existence and uniqueness of joint state extensions in Fermion systems, including cases with pure and mixed subsystem states.

Findings

Product state extension exists if all but one subsystem state are even.

Unique extension is guaranteed when all subsystem states are pure and conditions are met.

Non-uniqueness occurs in certain cases with non-pure subsystem states.

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.