Generalized Poincaré inequality for quantum Markov semigroups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912812374163456 |
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| author | Junge, Marius Wang, Jia |
| author_facet | Junge, Marius Wang, Jia |
| contents | We prove a noncommutative $(p,p)$-Poincaré inequality for trace-symmetric quantum Markov semigroups on tracial von Neumann algebras, assuming only the existence of a spectral gap. Extending semi-commutative results of Huang and Tropp, our argument uses Markov dilations to obtain chain-rule estimates for Dirichlet forms and employs amalgamated free products to define an appropriate noncommutative derivation. We further generalize the argument to non-tracial $σ$-finite von Neumann algebras under the weaker assumption of GNS-detailed balance, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover noncommutative Khintchine and sub-exponential concentration inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06005 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized Poincaré inequality for quantum Markov semigroups Junge, Marius Wang, Jia Operator Algebras Probability Quantum Physics We prove a noncommutative $(p,p)$-Poincaré inequality for trace-symmetric quantum Markov semigroups on tracial von Neumann algebras, assuming only the existence of a spectral gap. Extending semi-commutative results of Huang and Tropp, our argument uses Markov dilations to obtain chain-rule estimates for Dirichlet forms and employs amalgamated free products to define an appropriate noncommutative derivation. We further generalize the argument to non-tracial $σ$-finite von Neumann algebras under the weaker assumption of GNS-detailed balance, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover noncommutative Khintchine and sub-exponential concentration inequalities. |
| title | Generalized Poincaré inequality for quantum Markov semigroups |
| topic | Operator Algebras Probability Quantum Physics |
| url | https://arxiv.org/abs/2601.06005 |