### Abstract

We first determine the order automorphisms of the set of all positive definite operators with respect to the usual order and to the so-called chaotic order. We then apply those results to the following problems: (1) description of all bijective transformations on the space of nonsingular density operators (quantum states) which preserve the Umegaki or the Belavkin-Staszewski relative entropy; (2) characterization of the logarithmic product as the essentially unique binary operation on the set of positive definite operators that makes it an ordered commutative group with respect to the chaotic order.

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

Pages (from-to) | 2158-2169 |

Number of pages | 12 |

Journal | Linear Algebra and Its Applications |

Volume | 434 |

Issue number | 10 |

DOIs | |

Publication status | Published - May 15 2011 |

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### Keywords

- Chaotic order
- Logarithmic product
- Order
- Order automorphisms
- Ordered groups
- Positive definite operators
- Preservers
- Quantum relative entropies

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Numerical Analysis

### Cite this

**Order automorphisms on positive definite operators and a few applications.** / Molnár, L.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 434, no. 10, pp. 2158-2169. https://doi.org/10.1016/j.laa.2010.12.007

}

TY - JOUR

T1 - Order automorphisms on positive definite operators and a few applications

AU - Molnár, L.

PY - 2011/5/15

Y1 - 2011/5/15

N2 - We first determine the order automorphisms of the set of all positive definite operators with respect to the usual order and to the so-called chaotic order. We then apply those results to the following problems: (1) description of all bijective transformations on the space of nonsingular density operators (quantum states) which preserve the Umegaki or the Belavkin-Staszewski relative entropy; (2) characterization of the logarithmic product as the essentially unique binary operation on the set of positive definite operators that makes it an ordered commutative group with respect to the chaotic order.

AB - We first determine the order automorphisms of the set of all positive definite operators with respect to the usual order and to the so-called chaotic order. We then apply those results to the following problems: (1) description of all bijective transformations on the space of nonsingular density operators (quantum states) which preserve the Umegaki or the Belavkin-Staszewski relative entropy; (2) characterization of the logarithmic product as the essentially unique binary operation on the set of positive definite operators that makes it an ordered commutative group with respect to the chaotic order.

KW - Chaotic order

KW - Logarithmic product

KW - Order

KW - Order automorphisms

KW - Ordered groups

KW - Positive definite operators

KW - Preservers

KW - Quantum relative entropies

UR - http://www.scopus.com/inward/record.url?scp=79951681427&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79951681427&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2010.12.007

DO - 10.1016/j.laa.2010.12.007

M3 - Article

AN - SCOPUS:79951681427

VL - 434

SP - 2158

EP - 2169

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 10

ER -