---
_id: '12214'
abstract:
- lang: eng
  text: 'Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein
    space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0
    < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that
    Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is
    a consequence of our more general result: we prove that W1(X) is isometrically
    rigid if X is a complete separable metric space that satisfies the strict triangle
    inequality. Furthermore, we show that this latter rigidity result does not generalise
    to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence
    of mass-splitting isometries. '
acknowledgement: "Geher was supported by the Leverhulme Trust Early Career Fellowship
  (ECF-2018-125), and also by the Hungarian National Research, Development and Innovation
  Office - NKFIH (grant no. K115383 and K134944).\r\nTitkos was supported by the Hungarian
  National Research, Development and Innovation Office - NKFIH (grant no. PD128374,
  grant no. K115383 and K134944), by the J´anos Bolyai Research Scholarship of the
  Hungarian Academy of Sciences, and by the UNKP-20-5-BGE-1 New National Excellence
  Program of the ´Ministry of Innovation and Technology.\r\nVirosztek was supported
  by the European Union’s Horizon 2020 research and innovation program under the Marie
  Sklodowska-Curie Grant Agreement No. 846294, by the Momentum program of the Hungarian
  Academy of Sciences under grant agreement no. LP2021-15/2021, and partially supported
  by the Hungarian National Research, Development and Innovation Office - NKFIH (grants
  no. K124152 and no. KH129601). "
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: György Pál
  full_name: Gehér, György Pál
  last_name: Gehér
- first_name: Tamás
  full_name: Titkos, Tamás
  last_name: Titkos
- first_name: Daniel
  full_name: Virosztek, Daniel
  id: 48DB45DA-F248-11E8-B48F-1D18A9856A87
  last_name: Virosztek
  orcid: 0000-0003-1109-5511
citation:
  ama: 'Gehér GP, Titkos T, Virosztek D. The isometry group of Wasserstein spaces:
    The Hilbertian case. <i>Journal of the London Mathematical Society</i>. 2022;106(4):3865-3894.
    doi:<a href="https://doi.org/10.1112/jlms.12676">10.1112/jlms.12676</a>'
  apa: 'Gehér, G. P., Titkos, T., &#38; Virosztek, D. (2022). The isometry group of
    Wasserstein spaces: The Hilbertian case. <i>Journal of the London Mathematical
    Society</i>. Wiley. <a href="https://doi.org/10.1112/jlms.12676">https://doi.org/10.1112/jlms.12676</a>'
  chicago: 'Gehér, György Pál, Tamás Titkos, and Daniel Virosztek. “The Isometry Group
    of Wasserstein Spaces: The Hilbertian Case.” <i>Journal of the London Mathematical
    Society</i>. Wiley, 2022. <a href="https://doi.org/10.1112/jlms.12676">https://doi.org/10.1112/jlms.12676</a>.'
  ieee: 'G. P. Gehér, T. Titkos, and D. Virosztek, “The isometry group of Wasserstein
    spaces: The Hilbertian case,” <i>Journal of the London Mathematical Society</i>,
    vol. 106, no. 4. Wiley, pp. 3865–3894, 2022.'
  ista: 'Gehér GP, Titkos T, Virosztek D. 2022. The isometry group of Wasserstein
    spaces: The Hilbertian case. Journal of the London Mathematical Society. 106(4),
    3865–3894.'
  mla: 'Gehér, György Pál, et al. “The Isometry Group of Wasserstein Spaces: The Hilbertian
    Case.” <i>Journal of the London Mathematical Society</i>, vol. 106, no. 4, Wiley,
    2022, pp. 3865–94, doi:<a href="https://doi.org/10.1112/jlms.12676">10.1112/jlms.12676</a>.'
  short: G.P. Gehér, T. Titkos, D. Virosztek, Journal of the London Mathematical Society
    106 (2022) 3865–3894.
date_created: 2023-01-16T09:46:13Z
date_published: 2022-09-18T00:00:00Z
date_updated: 2023-08-04T09:24:17Z
day: '18'
department:
- _id: LaEr
doi: 10.1112/jlms.12676
ec_funded: 1
external_id:
  arxiv:
  - '2102.02037'
  isi:
  - '000854878500001'
intvolume: '       106'
isi: 1
issue: '4'
keyword:
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2102.02037
month: '09'
oa: 1
oa_version: Preprint
page: 3865-3894
project:
- _id: 26A455A6-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '846294'
  name: Geometric study of Wasserstein spaces and free probability
publication: Journal of the London Mathematical Society
publication_identifier:
  eissn:
  - 1469-7750
  issn:
  - 0024-6107
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'The isometry group of Wasserstein spaces: The Hilbertian case'
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 106
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'
...
---
_id: '9036'
abstract:
- lang: eng
  text: In this short note, we prove that the square root of the quantum Jensen-Shannon
    divergence is a true metric on the cone of positive matrices, and hence in particular
    on the quantum state space.
acknowledgement: D. Virosztek was supported by the European Union's Horizon 2020 research
  and innovation programme under the Marie Skłodowska-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_number: '107595'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Daniel
  full_name: Virosztek, Daniel
  id: 48DB45DA-F248-11E8-B48F-1D18A9856A87
  last_name: Virosztek
  orcid: 0000-0003-1109-5511
citation:
  ama: Virosztek D. The metric property of the quantum Jensen-Shannon divergence.
    <i>Advances in Mathematics</i>. 2021;380(3). doi:<a href="https://doi.org/10.1016/j.aim.2021.107595">10.1016/j.aim.2021.107595</a>
  apa: Virosztek, D. (2021). The metric property of the quantum Jensen-Shannon divergence.
    <i>Advances in Mathematics</i>. Elsevier. <a href="https://doi.org/10.1016/j.aim.2021.107595">https://doi.org/10.1016/j.aim.2021.107595</a>
  chicago: Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.”
    <i>Advances in Mathematics</i>. Elsevier, 2021. <a href="https://doi.org/10.1016/j.aim.2021.107595">https://doi.org/10.1016/j.aim.2021.107595</a>.
  ieee: D. Virosztek, “The metric property of the quantum Jensen-Shannon divergence,”
    <i>Advances in Mathematics</i>, vol. 380, no. 3. Elsevier, 2021.
  ista: Virosztek D. 2021. The metric property of the quantum Jensen-Shannon divergence.
    Advances in Mathematics. 380(3), 107595.
  mla: Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.”
    <i>Advances in Mathematics</i>, vol. 380, no. 3, 107595, Elsevier, 2021, doi:<a
    href="https://doi.org/10.1016/j.aim.2021.107595">10.1016/j.aim.2021.107595</a>.
  short: D. Virosztek, Advances in Mathematics 380 (2021).
date_created: 2021-01-22T17:55:17Z
date_published: 2021-03-26T00:00:00Z
date_updated: 2023-08-07T13:34:48Z
day: '26'
department:
- _id: LaEr
doi: 10.1016/j.aim.2021.107595
ec_funded: 1
external_id:
  arxiv:
  - '1910.10447'
  isi:
  - '000619676100035'
intvolume: '       380'
isi: 1
issue: '3'
keyword:
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1910.10447
month: '03'
oa: 1
oa_version: Preprint
project:
- _id: 26A455A6-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '846294'
  name: Geometric study of Wasserstein spaces and free probability
publication: Advances in Mathematics
publication_identifier:
  issn:
  - 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: The metric property of the quantum Jensen-Shannon divergence
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 380
year: '2021'
...
---
_id: '7389'
abstract:
- lang: eng
  text: "Recently Kloeckner described the structure of the isometry group of the quadratic
    Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional
    in the sense that there exists an exotic isometry flow. Following this line of
    investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein
    space\r\nW_p(R) for all p \\in [1,\\infty) \\setminus {2}. We show that W_2(R)
    is also exceptional regarding the\r\nparameter p: W_p(R) is isometrically rigid
    if and only if p is not equal to 2. Regarding the underlying\r\nspace, we prove
    that the exceptionality of p = 2 disappears if we replace R by the compact\r\ninterval
    [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only
    if\r\np is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass,
    and Isom(W_1([0,1]))\r\ncannot be embedded into Isom(W_1(R))."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Gyorgy Pal
  full_name: Geher, Gyorgy Pal
  last_name: Geher
- first_name: Tamas
  full_name: Titkos, Tamas
  last_name: Titkos
- first_name: Daniel
  full_name: Virosztek, Daniel
  id: 48DB45DA-F248-11E8-B48F-1D18A9856A87
  last_name: Virosztek
  orcid: 0000-0003-1109-5511
citation:
  ama: Geher GP, Titkos T, Virosztek D. Isometric study of Wasserstein spaces - the
    real line. <i>Transactions of the American Mathematical Society</i>. 2020;373(8):5855-5883.
    doi:<a href="https://doi.org/10.1090/tran/8113">10.1090/tran/8113</a>
  apa: Geher, G. P., Titkos, T., &#38; Virosztek, D. (2020). Isometric study of Wasserstein
    spaces - the real line. <i>Transactions of the American Mathematical Society</i>.
    American Mathematical Society. <a href="https://doi.org/10.1090/tran/8113">https://doi.org/10.1090/tran/8113</a>
  chicago: Geher, Gyorgy Pal, Tamas Titkos, and Daniel Virosztek. “Isometric Study
    of Wasserstein Spaces - the Real Line.” <i>Transactions of the American Mathematical
    Society</i>. American Mathematical Society, 2020. <a href="https://doi.org/10.1090/tran/8113">https://doi.org/10.1090/tran/8113</a>.
  ieee: G. P. Geher, T. Titkos, and D. Virosztek, “Isometric study of Wasserstein
    spaces - the real line,” <i>Transactions of the American Mathematical Society</i>,
    vol. 373, no. 8. American Mathematical Society, pp. 5855–5883, 2020.
  ista: Geher GP, Titkos T, Virosztek D. 2020. Isometric study of Wasserstein spaces
    - the real line. Transactions of the American Mathematical Society. 373(8), 5855–5883.
  mla: Geher, Gyorgy Pal, et al. “Isometric Study of Wasserstein Spaces - the Real
    Line.” <i>Transactions of the American Mathematical Society</i>, vol. 373, no.
    8, American Mathematical Society, 2020, pp. 5855–83, doi:<a href="https://doi.org/10.1090/tran/8113">10.1090/tran/8113</a>.
  short: G.P. Geher, T. Titkos, D. Virosztek, Transactions of the American Mathematical
    Society 373 (2020) 5855–5883.
date_created: 2020-01-29T10:20:46Z
date_published: 2020-08-01T00:00:00Z
date_updated: 2023-08-17T14:31:03Z
day: '01'
ddc:
- '515'
department:
- _id: LaEr
doi: 10.1090/tran/8113
ec_funded: 1
external_id:
  arxiv:
  - '2002.00859'
  isi:
  - '000551418100018'
intvolume: '       373'
isi: 1
issue: '8'
keyword:
- Wasserstein space
- isometric embeddings
- isometric rigidity
- exotic isometry flow
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2002.00859
month: '08'
oa: 1
oa_version: Preprint
page: 5855-5883
project:
- _id: 26A455A6-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '846294'
  name: Geometric study of Wasserstein spaces and free probability
publication: Transactions of the American Mathematical Society
publication_identifier:
  eissn:
  - '10886850'
  issn:
  - '00029947'
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: Isometric study of Wasserstein spaces - the real line
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 373
year: '2020'
...
---
_id: '7618'
abstract:
- lang: eng
  text: 'This short note aims to study quantum Hellinger distances investigated recently
    by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis
    on barycenters. We introduce the family of generalized quantum Hellinger divergences
    that are of the form ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando
    mean, and c∈(0,1) is the weight of σ. We note that these divergences belong to
    the family of maximal quantum f-divergences, and hence are jointly convex, and
    satisfy the data processing inequality. We derive a characterization of the barycenter
    of finitely many positive definite operators for these generalized quantum Hellinger
    divergences. We note that the characterization of the barycenter as the weighted
    multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true
    in the case of commuting operators, but it is not correct in the general case. '
acknowledgement: "J. Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum
  Grant for Quantum\r\nInformation Theory, No. 96 141, and by the Hungarian National
  Research, Development and Innovation\r\nOffice (NKFIH) via Grants Nos. K119442,
  K124152 and KH129601. D. Virosztek was supported by the\r\nISTFELLOW program of
  the Institute of Science and Technology Austria (Project Code IC1027FELL01),\r\nby
  the European Union’s Horizon 2020 research and innovation program under the Marie\r\nSklodowska-Curie
  Grant Agreement No. 846294, and partially supported by the Hungarian National\r\nResearch,
  Development and Innovation Office (NKFIH) via Grants Nos. K124152 and KH129601.\r\nWe
  are grateful to Milán Mosonyi for drawing our attention to Ref.’s [6,14,15,17,\r\n20,21],
  for comments on earlier versions of this paper, and for several discussions on the
  topic. We are\r\nalso grateful to Miklós Pálfia for several discussions; to László
  Erdös for his essential suggestions on the\r\nstructure and highlights of this paper,
  and for his comments on earlier versions; and to the anonymous\r\nreferee for his/her
  valuable comments and suggestions."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jozsef
  full_name: Pitrik, Jozsef
  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. Quantum Hellinger distances revisited. <i>Letters in
    Mathematical Physics</i>. 2020;110(8):2039-2052. doi:<a href="https://doi.org/10.1007/s11005-020-01282-0">10.1007/s11005-020-01282-0</a>
  apa: Pitrik, J., &#38; Virosztek, D. (2020). Quantum Hellinger distances revisited.
    <i>Letters in Mathematical Physics</i>. Springer Nature. <a href="https://doi.org/10.1007/s11005-020-01282-0">https://doi.org/10.1007/s11005-020-01282-0</a>
  chicago: Pitrik, Jozsef, and Daniel Virosztek. “Quantum Hellinger Distances Revisited.”
    <i>Letters in Mathematical Physics</i>. Springer Nature, 2020. <a href="https://doi.org/10.1007/s11005-020-01282-0">https://doi.org/10.1007/s11005-020-01282-0</a>.
  ieee: J. Pitrik and D. Virosztek, “Quantum Hellinger distances revisited,” <i>Letters
    in Mathematical Physics</i>, vol. 110, no. 8. Springer Nature, pp. 2039–2052,
    2020.
  ista: Pitrik J, Virosztek D. 2020. Quantum Hellinger distances revisited. Letters
    in Mathematical Physics. 110(8), 2039–2052.
  mla: Pitrik, Jozsef, and Daniel Virosztek. “Quantum Hellinger Distances Revisited.”
    <i>Letters in Mathematical Physics</i>, vol. 110, no. 8, Springer Nature, 2020,
    pp. 2039–52, doi:<a href="https://doi.org/10.1007/s11005-020-01282-0">10.1007/s11005-020-01282-0</a>.
  short: J. Pitrik, D. Virosztek, Letters in Mathematical Physics 110 (2020) 2039–2052.
date_created: 2020-03-25T15:57:48Z
date_published: 2020-08-01T00:00:00Z
date_updated: 2023-08-18T10:17:26Z
day: '01'
department:
- _id: LaEr
doi: 10.1007/s11005-020-01282-0
ec_funded: 1
external_id:
  arxiv:
  - '1903.10455'
  isi:
  - '000551556000002'
intvolume: '       110'
isi: 1
issue: '8'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1903.10455
month: '08'
oa: 1
oa_version: Preprint
page: 2039-2052
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: Letters in Mathematical Physics
publication_identifier:
  eissn:
  - 1573-0530
  issn:
  - 0377-9017
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Quantum Hellinger distances revisited
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 110
year: '2020'
...