### Abstract

Given a positive function f on (0, ∞) and a non-zero real parameter θ, we consider a function I^{θ}_{f}(A,B,X) = tr X^{*}(f(L_{A}R^{-1}_{B})R_{B})^{-θ}(X)in three matrices A, B > 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of I^{θ}_{f}(A,B,X). Similar discussions are given for quasi-entropies and quantum skew informations.

Original language | English |
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Article number | 062201 |

Journal | Journal of Mathematical Physics |

Volume | 54 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 3 2013 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Convexity of quasi-entropy type functions : Lieb's and Ando's convexity theorems revisited.** / Hiai, Fumio; Petz, D.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 54, no. 6, 062201. https://doi.org/10.1063/1.4810781

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TY - JOUR

T1 - Convexity of quasi-entropy type functions

T2 - Lieb's and Ando's convexity theorems revisited

AU - Hiai, Fumio

AU - Petz, D.

PY - 2013/6/3

Y1 - 2013/6/3

N2 - Given a positive function f on (0, ∞) and a non-zero real parameter θ, we consider a function Iθf(A,B,X) = tr X*(f(LAR-1B)RB)-θ(X)in three matrices A, B > 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of Iθf(A,B,X). Similar discussions are given for quasi-entropies and quantum skew informations.

AB - Given a positive function f on (0, ∞) and a non-zero real parameter θ, we consider a function Iθf(A,B,X) = tr X*(f(LAR-1B)RB)-θ(X)in three matrices A, B > 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of Iθf(A,B,X). Similar discussions are given for quasi-entropies and quantum skew informations.

UR - http://www.scopus.com/inward/record.url?scp=84880120440&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880120440&partnerID=8YFLogxK

U2 - 10.1063/1.4810781

DO - 10.1063/1.4810781

M3 - Article

AN - SCOPUS:84880120440

VL - 54

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 6

M1 - 062201

ER -