### 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.

Original language | English |
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Pages (from-to) | 105-122 |

Number of pages | 18 |

Journal | Communications in Mathematical Physics |

Volume | 237 |

Issue number | 1-2 |

Publication status | Published - 2003 Jun |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Communications in Mathematical Physics*,

*237*(1-2), 105-122.

**Joint extension of states of subsystems for a CAR system.** / Araki, Huzihiro; Moriya, Hajime.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 237, no. 1-2, pp. 105-122.

}

TY - JOUR

T1 - Joint extension of states of subsystems for a CAR system

AU - Araki, Huzihiro

AU - Moriya, Hajime

PY - 2003/6

Y1 - 2003/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0037972656&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037972656&partnerID=8YFLogxK

M3 - Article

VL - 237

SP - 105

EP - 122

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1-2

ER -