### Abstract

Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. Two subalgebras A_{1} and A_{2} of B(H) are called complementary if the traceless subspaces of A_{1} and A_{2} are orthogonal (with respect to the Hilbert Schmidt inner product). When both subalgebras are maximal Abelian, then the concept reduces to complementary observables or mutually unbiased bases. In the paper several characterizations of complementary subalgebras are given in the general case and several examples are presented. For a 4-level quantum system, the structure of complementary subalgebras can be described very well, the Cartan decomposition of unitaries plays a role. It turns out that a measurement corresponding to the Bell basis is complementary to any local measurement of the two-qubit system.

Original language | English |
---|---|

Pages (from-to) | 209-224 |

Number of pages | 16 |

Journal | Reports on Mathematical Physics |

Volume | 59 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2007 |

### Fingerprint

### Keywords

- Bell states
- CAR algebra
- Cartan decomposition
- commuting squares
- complementarity
- entropic uncertainty relation
- mutually unbiased basis

### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

**Complementarity in quantum systems.** / Petz, D.

Research output: Contribution to journal › Article

*Reports on Mathematical Physics*, vol. 59, no. 2, pp. 209-224. https://doi.org/10.1016/S0034-4877(07)00010-9

}

TY - JOUR

T1 - Complementarity in quantum systems

AU - Petz, D.

PY - 2007/4

Y1 - 2007/4

N2 - Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. Two subalgebras A1 and A2 of B(H) are called complementary if the traceless subspaces of A1 and A2 are orthogonal (with respect to the Hilbert Schmidt inner product). When both subalgebras are maximal Abelian, then the concept reduces to complementary observables or mutually unbiased bases. In the paper several characterizations of complementary subalgebras are given in the general case and several examples are presented. For a 4-level quantum system, the structure of complementary subalgebras can be described very well, the Cartan decomposition of unitaries plays a role. It turns out that a measurement corresponding to the Bell basis is complementary to any local measurement of the two-qubit system.

AB - Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. Two subalgebras A1 and A2 of B(H) are called complementary if the traceless subspaces of A1 and A2 are orthogonal (with respect to the Hilbert Schmidt inner product). When both subalgebras are maximal Abelian, then the concept reduces to complementary observables or mutually unbiased bases. In the paper several characterizations of complementary subalgebras are given in the general case and several examples are presented. For a 4-level quantum system, the structure of complementary subalgebras can be described very well, the Cartan decomposition of unitaries plays a role. It turns out that a measurement corresponding to the Bell basis is complementary to any local measurement of the two-qubit system.

KW - Bell states

KW - CAR algebra

KW - Cartan decomposition

KW - commuting squares

KW - complementarity

KW - entropic uncertainty relation

KW - mutually unbiased basis

UR - http://www.scopus.com/inward/record.url?scp=34249713440&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249713440&partnerID=8YFLogxK

U2 - 10.1016/S0034-4877(07)00010-9

DO - 10.1016/S0034-4877(07)00010-9

M3 - Article

AN - SCOPUS:34249713440

VL - 59

SP - 209

EP - 224

JO - Reports on Mathematical Physics

JF - Reports on Mathematical Physics

SN - 0034-4877

IS - 2

ER -