---
_id: '12216'
abstract:
- lang: eng
  text: Many trace inequalities can be expressed either as concavity/convexity theorems
    or as monotonicity theorems. A classic example is the joint convexity of the quantum
    relative entropy which is equivalent to the Data Processing Inequality. The latter
    says that quantum operations can never increase the relative entropy. The monotonicity
    versions often have many advantages, and often have direct physical application,
    as in the example just mentioned. Moreover, the monotonicity results are often
    valid for a larger class of maps than, say, quantum operations (which are completely
    positive). In this paper we prove several new monotonicity results, the first
    of which is a monotonicity theorem that has as a simple corollary a celebrated
    concavity theorem of Epstein. Our starting points are the monotonicity versions
    of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs
    of these in their general forms using interpolation. We then prove our new monotonicity
    theorems by several duality arguments.
acknowledgement: Work partially supported by the Lise Meitner fellowship, Austrian
  Science Fund (FWF) M3337.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Eric A.
  full_name: Carlen, Eric A.
  last_name: Carlen
- first_name: Haonan
  full_name: Zhang, Haonan
  id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
  last_name: Zhang
citation:
  ama: Carlen EA, Zhang H. Monotonicity versions of Epstein’s concavity theorem and
    related inequalities. <i>Linear Algebra and its Applications</i>. 2022;654:289-310.
    doi:<a href="https://doi.org/10.1016/j.laa.2022.09.001">10.1016/j.laa.2022.09.001</a>
  apa: Carlen, E. A., &#38; Zhang, H. (2022). Monotonicity versions of Epstein’s concavity
    theorem and related inequalities. <i>Linear Algebra and Its Applications</i>.
    Elsevier. <a href="https://doi.org/10.1016/j.laa.2022.09.001">https://doi.org/10.1016/j.laa.2022.09.001</a>
  chicago: Carlen, Eric A., and Haonan Zhang. “Monotonicity Versions of Epstein’s
    Concavity Theorem and Related Inequalities.” <i>Linear Algebra and Its Applications</i>.
    Elsevier, 2022. <a href="https://doi.org/10.1016/j.laa.2022.09.001">https://doi.org/10.1016/j.laa.2022.09.001</a>.
  ieee: E. A. Carlen and H. Zhang, “Monotonicity versions of Epstein’s concavity theorem
    and related inequalities,” <i>Linear Algebra and its Applications</i>, vol. 654.
    Elsevier, pp. 289–310, 2022.
  ista: Carlen EA, Zhang H. 2022. Monotonicity versions of Epstein’s concavity theorem
    and related inequalities. Linear Algebra and its Applications. 654, 289–310.
  mla: Carlen, Eric A., and Haonan Zhang. “Monotonicity Versions of Epstein’s Concavity
    Theorem and Related Inequalities.” <i>Linear Algebra and Its Applications</i>,
    vol. 654, Elsevier, 2022, pp. 289–310, doi:<a href="https://doi.org/10.1016/j.laa.2022.09.001">10.1016/j.laa.2022.09.001</a>.
  short: E.A. Carlen, H. Zhang, Linear Algebra and Its Applications 654 (2022) 289–310.
date_created: 2023-01-16T09:46:38Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-04T09:24:51Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1016/j.laa.2022.09.001
external_id:
  isi:
  - '000860689600014'
file:
- access_level: open_access
  checksum: cf3cb7e7e34baa967849f01d8f0c1ae4
  content_type: application/pdf
  creator: dernst
  date_created: 2023-01-27T08:08:39Z
  date_updated: 2023-01-27T08:08:39Z
  file_id: '12415'
  file_name: 2022_LinearAlgebra_Carlen.pdf
  file_size: 441184
  relation: main_file
  success: 1
file_date_updated: 2023-01-27T08:08:39Z
has_accepted_license: '1'
intvolume: '       654'
isi: 1
keyword:
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Numerical Analysis
- Algebra and Number Theory
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '12'
oa: 1
oa_version: Published Version
page: 289-310
project:
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
  grant_number: M03337
  name: Curvature-dimension in noncommutative analysis
publication: Linear Algebra and its Applications
publication_identifier:
  issn:
  - 0024-3795
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Monotonicity versions of Epstein's concavity theorem and related inequalities
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 654
year: '2022'
...
---
_id: '8373'
abstract:
- lang: eng
  text: It is well known that special Kubo-Ando operator means admit divergence center
    interpretations, moreover, they are also mean squared error estimators for certain
    metrics on positive definite operators. In this paper we give a divergence center
    interpretation for every symmetric Kubo-Ando mean. This characterization of the
    symmetric means naturally leads to a definition of weighted and multivariate versions
    of a large class of symmetric Kubo-Ando means. We study elementary properties
    of these weighted multivariate means, and note in particular that in the special
    case of the geometric mean we recover the weighted A#H-mean introduced by Kim,
    Lawson, and Lim.
acknowledgement: "The authors are grateful to Milán Mosonyi for fruitful discussions
  on the topic, and to the anonymous referee for his/her comments and suggestions.\r\nJ.
  Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant
  for Quantum Information Theory, No. 96 141, and by Hungarian National Research,
  Development and Innovation Office (NKFIH) via grants no. K119442, no. K124152, and
  no. KH129601. D. Virosztek was supported by the ISTFELLOW program of the Institute
  of Science and Technology Austria (project code IC1027FELL01), by the European Union's
  Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant
  Agreement No. 846294, and partially supported by the Hungarian National Research,
  Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: József
  full_name: Pitrik, József
  last_name: Pitrik
- first_name: Daniel
  full_name: Virosztek, Daniel
  id: 48DB45DA-F248-11E8-B48F-1D18A9856A87
  last_name: Virosztek
  orcid: 0000-0003-1109-5511
citation:
  ama: Pitrik J, Virosztek D. A divergence center interpretation of general symmetric
    Kubo-Ando means, and related weighted multivariate operator means. <i>Linear Algebra
    and its Applications</i>. 2021;609:203-217. doi:<a href="https://doi.org/10.1016/j.laa.2020.09.007">10.1016/j.laa.2020.09.007</a>
  apa: Pitrik, J., &#38; Virosztek, D. (2021). A divergence center interpretation
    of general symmetric Kubo-Ando means, and related weighted multivariate operator
    means. <i>Linear Algebra and Its Applications</i>. Elsevier. <a href="https://doi.org/10.1016/j.laa.2020.09.007">https://doi.org/10.1016/j.laa.2020.09.007</a>
  chicago: Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation
    of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator
    Means.” <i>Linear Algebra and Its Applications</i>. Elsevier, 2021. <a href="https://doi.org/10.1016/j.laa.2020.09.007">https://doi.org/10.1016/j.laa.2020.09.007</a>.
  ieee: J. Pitrik and D. Virosztek, “A divergence center interpretation of general
    symmetric Kubo-Ando means, and related weighted multivariate operator means,”
    <i>Linear Algebra and its Applications</i>, vol. 609. Elsevier, pp. 203–217, 2021.
  ista: Pitrik J, Virosztek D. 2021. A divergence center interpretation of general
    symmetric Kubo-Ando means, and related weighted multivariate operator means. Linear
    Algebra and its Applications. 609, 203–217.
  mla: Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of
    General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator
    Means.” <i>Linear Algebra and Its Applications</i>, vol. 609, Elsevier, 2021,
    pp. 203–17, doi:<a href="https://doi.org/10.1016/j.laa.2020.09.007">10.1016/j.laa.2020.09.007</a>.
  short: J. Pitrik, D. Virosztek, Linear Algebra and Its Applications 609 (2021) 203–217.
date_created: 2020-09-11T08:35:50Z
date_published: 2021-01-15T00:00:00Z
date_updated: 2023-08-04T10:58:14Z
day: '15'
department:
- _id: LaEr
doi: 10.1016/j.laa.2020.09.007
ec_funded: 1
external_id:
  arxiv:
  - '2002.11678'
  isi:
  - '000581730500011'
intvolume: '       609'
isi: 1
keyword:
- Kubo-Ando mean
- weighted multivariate mean
- barycenter
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2002.11678
month: '01'
oa: 1
oa_version: Preprint
page: 203-217
project:
- _id: 26A455A6-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '846294'
  name: Geometric study of Wasserstein spaces and free probability
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
publication: Linear Algebra and its Applications
publication_identifier:
  issn:
  - 0024-3795
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: A divergence center interpretation of general symmetric Kubo-Ando means, and
  related weighted multivariate operator means
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 609
year: '2021'
...