---
_id: '13271'
abstract:
- lang: eng
text: "In this paper, we prove the convexity of trace functionals (A,B,C)↦Tr|BpACq|s,\r\nfor
parameters (p, q, s) that are best possible, where B and C are any n-by-n positive-definite
matrices, and A is any n-by-n matrix. We also obtain the monotonicity versions
of trace functionals of this type. As applications, we extend some results in
Carlen et al. (Linear Algebra Appl 490:174–185, 2016), Hiai and Petz (Publ Res
Inst Math Sci 48(3):525-542, 2012) and resolve a conjecture in Al-Rashed and Zegarliński
(Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) in the matrix
setting. Other conjectures in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum
Probab Relat Top 17(4):1450029, 2014) will also be discussed. We also show that
some related trace functionals are not concave in general. Such concavity results
were expected to hold in different problems."
acknowledgement: I am grateful to Boguslaw Zegarliński for asking me the questions
in [3] and for helpful communication. I also want to thank Paata Ivanisvili for
drawing [25] to my attention and for useful correspondence. Many thanks to the anonymous
referee for the valuable comments and for pointing out some errors in an earlier
version of the paper. This work is partially supported by the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement
No. 754411 and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Haonan
full_name: Zhang, Haonan
id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
last_name: Zhang
citation:
ama: Zhang H. Some convexity and monotonicity results of trace functionals. Annales
Henri Poincare. 2023. doi:10.1007/s00023-023-01345-7
apa: Zhang, H. (2023). Some convexity and monotonicity results of trace functionals.
Annales Henri Poincare. Springer Nature. https://doi.org/10.1007/s00023-023-01345-7
chicago: Zhang, Haonan. “Some Convexity and Monotonicity Results of Trace Functionals.”
Annales Henri Poincare. Springer Nature, 2023. https://doi.org/10.1007/s00023-023-01345-7.
ieee: H. Zhang, “Some convexity and monotonicity results of trace functionals,”
Annales Henri Poincare. Springer Nature, 2023.
ista: Zhang H. 2023. Some convexity and monotonicity results of trace functionals.
Annales Henri Poincare.
mla: Zhang, Haonan. “Some Convexity and Monotonicity Results of Trace Functionals.”
Annales Henri Poincare, Springer Nature, 2023, doi:10.1007/s00023-023-01345-7.
short: H. Zhang, Annales Henri Poincare (2023).
date_created: 2023-07-23T22:01:15Z
date_published: 2023-07-08T00:00:00Z
date_updated: 2023-12-13T11:33:46Z
day: '08'
department:
- _id: JaMa
doi: 10.1007/s00023-023-01345-7
ec_funded: 1
external_id:
arxiv:
- '2108.05785'
isi:
- '001025709100001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2108.05785
month: '07'
oa: 1
oa_version: Preprint
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
grant_number: M03337
name: Curvature-dimension in noncommutative analysis
publication: Annales Henri Poincare
publication_identifier:
issn:
- 1424-0637
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Some convexity and monotonicity results of trace functionals
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '13318'
abstract:
- lang: eng
text: Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free
constants that grow subexponentially in the degree (Defant et al. in Math Ann
374(1):653–680, 2019). Such inequalities have found great applications in learning
low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th
annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated
by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality
for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand,
KKL and Friedgut’s theorems and the learnability of quantum Boolean functions,
2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et
al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894).
In this paper, we give a new proof of these Bohnenblust–Hille inequalities for
qubit system with constants that are dimension-free and of exponential growth
in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials.
Using similar ideas, we also study learning problems of low degree quantum observables
and Bohr’s radius phenomenon on quantum Boolean cubes.
acknowledgement: The research of A.V. is supported by NSF DMS-1900286, DMS-2154402
and by Hausdorff Center for Mathematics. H.Z. is supported by the Lise Meitner fellowship,
Austrian Science Fund (FWF) M3337. This work is partially supported by NSF DMS-1929284
while both authors were in residence at the Institute for Computational and Experimental
Research in Mathematics in Providence, RI, during the Harmonic Analysis and Convexity
program.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Alexander
full_name: Volberg, Alexander
last_name: Volberg
- first_name: Haonan
full_name: Zhang, Haonan
id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
last_name: Zhang
citation:
ama: Volberg A, Zhang H. Noncommutative Bohnenblust–Hille inequalities. Mathematische
Annalen. 2023. doi:10.1007/s00208-023-02680-0
apa: Volberg, A., & Zhang, H. (2023). Noncommutative Bohnenblust–Hille inequalities.
Mathematische Annalen. Springer Nature. https://doi.org/10.1007/s00208-023-02680-0
chicago: Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille
Inequalities.” Mathematische Annalen. Springer Nature, 2023. https://doi.org/10.1007/s00208-023-02680-0.
ieee: A. Volberg and H. Zhang, “Noncommutative Bohnenblust–Hille inequalities,”
Mathematische Annalen. Springer Nature, 2023.
ista: Volberg A, Zhang H. 2023. Noncommutative Bohnenblust–Hille inequalities. Mathematische
Annalen.
mla: Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille Inequalities.”
Mathematische Annalen, Springer Nature, 2023, doi:10.1007/s00208-023-02680-0.
short: A. Volberg, H. Zhang, Mathematische Annalen (2023).
date_created: 2023-07-30T22:01:03Z
date_published: 2023-07-24T00:00:00Z
date_updated: 2023-12-13T11:36:20Z
day: '24'
department:
- _id: JaMa
doi: 10.1007/s00208-023-02680-0
external_id:
arxiv:
- '2210.14468'
isi:
- '001035665500001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1007/s00208-023-02680-0
month: '07'
oa: 1
oa_version: Published Version
project:
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
grant_number: M03337
name: Curvature-dimension in noncommutative analysis
publication: Mathematische Annalen
publication_identifier:
eissn:
- 1432-1807
issn:
- 0025-5831
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Noncommutative Bohnenblust–Hille inequalities
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '12087'
abstract:
- lang: eng
text: Following up on the recent work on lower Ricci curvature bounds for quantum
systems, we introduce two noncommutative versions of curvature-dimension bounds
for symmetric quantum Markov semigroups over matrix algebras. Under suitable such
curvature-dimension conditions, we prove a family of dimension-dependent functional
inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power
in the noncommutative setting. We also provide examples satisfying certain curvature-dimension
conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers
over group algebras and generalized depolarizing semigroups.
acknowledgement: H.Z. is supported by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411
and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. M.W. acknowledges
support from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (Grant Agreement No. 716117) and from the
Austrian Science Fund (FWF) through grant number F65. Both authors would like to
thank Jan Maas for fruitful discussions and helpful comments. Open access funding
provided by Austrian Science Fund (FWF).
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Melchior
full_name: Wirth, Melchior
id: 88644358-0A0E-11EA-8FA5-49A33DDC885E
last_name: Wirth
orcid: 0000-0002-0519-4241
- first_name: Haonan
full_name: Zhang, Haonan
id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
last_name: Zhang
citation:
ama: Wirth M, Zhang H. Curvature-dimension conditions for symmetric quantum Markov
semigroups. Annales Henri Poincare. 2023;24:717-750. doi:10.1007/s00023-022-01220-x
apa: Wirth, M., & Zhang, H. (2023). Curvature-dimension conditions for symmetric
quantum Markov semigroups. Annales Henri Poincare. Springer Nature. https://doi.org/10.1007/s00023-022-01220-x
chicago: Wirth, Melchior, and Haonan Zhang. “Curvature-Dimension Conditions for
Symmetric Quantum Markov Semigroups.” Annales Henri Poincare. Springer
Nature, 2023. https://doi.org/10.1007/s00023-022-01220-x.
ieee: M. Wirth and H. Zhang, “Curvature-dimension conditions for symmetric quantum
Markov semigroups,” Annales Henri Poincare, vol. 24. Springer Nature, pp.
717–750, 2023.
ista: Wirth M, Zhang H. 2023. Curvature-dimension conditions for symmetric quantum
Markov semigroups. Annales Henri Poincare. 24, 717–750.
mla: Wirth, Melchior, and Haonan Zhang. “Curvature-Dimension Conditions for Symmetric
Quantum Markov Semigroups.” Annales Henri Poincare, vol. 24, Springer Nature,
2023, pp. 717–50, doi:10.1007/s00023-022-01220-x.
short: M. Wirth, H. Zhang, Annales Henri Poincare 24 (2023) 717–750.
date_created: 2022-09-11T22:01:57Z
date_published: 2023-03-01T00:00:00Z
date_updated: 2023-08-14T11:39:28Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00023-022-01220-x
ec_funded: 1
external_id:
arxiv:
- '2105.08303'
isi:
- '000837499800002'
file:
- access_level: open_access
checksum: 8c7b185eba5ccd92ef55c120f654222c
content_type: application/pdf
creator: dernst
date_created: 2023-08-14T11:38:28Z
date_updated: 2023-08-14T11:38:28Z
file_id: '14051'
file_name: 2023_AnnalesHenriPoincare_Wirth.pdf
file_size: 554871
relation: main_file
success: 1
file_date_updated: 2023-08-14T11:38:28Z
has_accepted_license: '1'
intvolume: ' 24'
isi: 1
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 717-750
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
grant_number: M03337
name: Curvature-dimension in noncommutative analysis
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '716117'
name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
grant_number: F6504
name: Taming Complexity in Partial Differential Systems
publication: Annales Henri Poincare
publication_identifier:
issn:
- 1424-0637
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Curvature-dimension conditions for symmetric quantum Markov semigroups
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2023'
...
---
_id: '12210'
abstract:
- lang: eng
text: "The aim of this paper is to find new estimates for the norms of functions
of a (minus) distinguished Laplace operator L on the ‘ax+b’ groups. The central
part is devoted to spectrally localized wave propagators, that is, functions of
the type ψ(L−−√)exp(itL−−√), with ψ∈C0(R). We show that for t→+∞, the convolution
kernel kt of this operator satisfies\r\n∥kt∥1≍t,∥kt∥∞≍1,\r\nso that the upper
estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary
component, we recall the Plancherel density of L and spend certain time presenting
and comparing different approaches to its calculation. Using its explicit form,
we estimate uniform norms of several functions of the shifted Laplace-Beltrami
operator Δ~, closely related to L. The functions include in particular exp(−tΔ~γ),
t>0,γ>0, and (Δ~−z)s, with complex z, s."
acknowledgement: "Yu. K. thanks Professor Waldemar Hebisch for valuable discussions
on the general context of multipliers on Lie groups. This work was started during
an ICL-CNRS fellowship of the second\r\nnamed author at the Imperial College London.
Yu. K. is supported by the ANR-19-CE40-0002 grant of the French National Research
Agency (ANR). H. Z. is supported by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411
and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. R. A. was supported
by the EPSRC grant EP/R003025. M. R. is supported by the EPSRC grant EP/R003025/2
and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rauan
full_name: Akylzhanov, Rauan
last_name: Akylzhanov
- first_name: Yulia
full_name: Kuznetsova, Yulia
last_name: Kuznetsova
- first_name: Michael
full_name: Ruzhansky, Michael
last_name: Ruzhansky
- first_name: Haonan
full_name: Zhang, Haonan
id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
last_name: Zhang
citation:
ama: Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. Norms of certain functions
of a distinguished Laplacian on the ax + b groups. Mathematische Zeitschrift.
2022;302(4):2327-2352. doi:10.1007/s00209-022-03143-z
apa: Akylzhanov, R., Kuznetsova, Y., Ruzhansky, M., & Zhang, H. (2022). Norms
of certain functions of a distinguished Laplacian on the ax + b groups. Mathematische
Zeitschrift. Springer Nature. https://doi.org/10.1007/s00209-022-03143-z
chicago: Akylzhanov, Rauan, Yulia Kuznetsova, Michael Ruzhansky, and Haonan Zhang.
“Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.”
Mathematische Zeitschrift. Springer Nature, 2022. https://doi.org/10.1007/s00209-022-03143-z.
ieee: R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, and H. Zhang, “Norms of certain
functions of a distinguished Laplacian on the ax + b groups,” Mathematische
Zeitschrift, vol. 302, no. 4. Springer Nature, pp. 2327–2352, 2022.
ista: Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. 2022. Norms of certain functions
of a distinguished Laplacian on the ax + b groups. Mathematische Zeitschrift.
302(4), 2327–2352.
mla: Akylzhanov, Rauan, et al. “Norms of Certain Functions of a Distinguished Laplacian
on the Ax + b Groups.” Mathematische Zeitschrift, vol. 302, no. 4, Springer
Nature, 2022, pp. 2327–52, doi:10.1007/s00209-022-03143-z.
short: R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, H. Zhang, Mathematische Zeitschrift
302 (2022) 2327–2352.
date_created: 2023-01-16T09:45:31Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-04T09:22:14Z
day: '01'
department:
- _id: JaMa
doi: 10.1007/s00209-022-03143-z
ec_funded: 1
external_id:
arxiv:
- '2101.00584'
isi:
- '000859680700001'
intvolume: ' 302'
isi: 1
issue: '4'
keyword:
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2101.00584
month: '12'
oa: 1
oa_version: Preprint
page: 2327-2352
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
grant_number: M03337
name: Curvature-dimension in noncommutative analysis
publication: Mathematische Zeitschrift
publication_identifier:
eissn:
- 1432-1823
issn:
- 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Norms of certain functions of a distinguished Laplacian on the ax + b groups
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 302
year: '2022'
...
---
_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. Linear Algebra and its Applications. 2022;654:289-310.
doi:10.1016/j.laa.2022.09.001
apa: Carlen, E. A., & Zhang, H. (2022). Monotonicity versions of Epstein’s concavity
theorem and related inequalities. Linear Algebra and Its Applications.
Elsevier. https://doi.org/10.1016/j.laa.2022.09.001
chicago: Carlen, Eric A., and Haonan Zhang. “Monotonicity Versions of Epstein’s
Concavity Theorem and Related Inequalities.” Linear Algebra and Its Applications.
Elsevier, 2022. https://doi.org/10.1016/j.laa.2022.09.001.
ieee: E. A. Carlen and H. Zhang, “Monotonicity versions of Epstein’s concavity theorem
and related inequalities,” Linear Algebra and its Applications, 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.” Linear Algebra and Its Applications,
vol. 654, Elsevier, 2022, pp. 289–310, doi:10.1016/j.laa.2022.09.001.
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
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'
...