Complete positivity order and relative entropy decay
We prove that for a GNS-symmetric quantum Markov semigroup, the complete modified logarithmic Sobolev constant is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this gives a short proof that every sub-Laplacian of a Hörmander system on a compact manif...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001178/type/journal_article |
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| author | Li Gao Marius Junge Nicholas LaRacuente Haojian Li |
| author_facet | Li Gao Marius Junge Nicholas LaRacuente Haojian Li |
| author_sort | Li Gao |
| collection | DOAJ |
| description | We prove that for a GNS-symmetric quantum Markov semigroup, the complete modified logarithmic Sobolev constant is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this gives a short proof that every sub-Laplacian of a Hörmander system on a compact manifold satisfies a modified log-Sobolev inequality uniformly for scalar and matrix-valued functions. For quantum Markov semigroups, we show that the complete modified logarithmic Sobolev constant is comparable to the spectral gap up to the logarithm of the dimension. Such estimates are asymptotically tight for a quantum birth-death process. Our results, along with the consequence of concentration inequalities, are applicable to GNS-symmetric semigroups on general von Neumann algebras. |
| format | Article |
| id | doaj-art-6dba559c35e54842ae0ce10e60dce644 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-6dba559c35e54842ae0ce10e60dce6442025-02-06T09:14:57ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.117Complete positivity order and relative entropy decayLi Gao0https://orcid.org/0000-0002-9220-0119Marius Junge1Nicholas LaRacuente2Haojian Li3School of Mathematics and Statistics Wuhan University, Wuhan, Hubei 430072, P.R.ChinaDepartment of Mathematics University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; E-mail:Department of Computer Science Indiana University, Bloomington, IN 47408, USA; E-mail:Zentrum Mathematik Technische Universität München, Garching, 85748, Germany; E-mail:We prove that for a GNS-symmetric quantum Markov semigroup, the complete modified logarithmic Sobolev constant is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this gives a short proof that every sub-Laplacian of a Hörmander system on a compact manifold satisfies a modified log-Sobolev inequality uniformly for scalar and matrix-valued functions. For quantum Markov semigroups, we show that the complete modified logarithmic Sobolev constant is comparable to the spectral gap up to the logarithm of the dimension. Such estimates are asymptotically tight for a quantum birth-death process. Our results, along with the consequence of concentration inequalities, are applicable to GNS-symmetric semigroups on general von Neumann algebras.https://www.cambridge.org/core/product/identifier/S2050509424001178/type/journal_article47D0746N5081P1739B62 |
| spellingShingle | Li Gao Marius Junge Nicholas LaRacuente Haojian Li Complete positivity order and relative entropy decay Forum of Mathematics, Sigma 47D07 46N50 81P17 39B62 |
| title | Complete positivity order and relative entropy decay |
| title_full | Complete positivity order and relative entropy decay |
| title_fullStr | Complete positivity order and relative entropy decay |
| title_full_unstemmed | Complete positivity order and relative entropy decay |
| title_short | Complete positivity order and relative entropy decay |
| title_sort | complete positivity order and relative entropy decay |
| topic | 47D07 46N50 81P17 39B62 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001178/type/journal_article |
| work_keys_str_mv | AT ligao completepositivityorderandrelativeentropydecay AT mariusjunge completepositivityorderandrelativeentropydecay AT nicholaslaracuente completepositivityorderandrelativeentropydecay AT haojianli completepositivityorderandrelativeentropydecay |