Quantum Talagrand-type Inequalities via Variance Decay
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917277404758016 |
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| author | Chang, Fan Li, Peijie |
| author_facet | Chang, Fan Li, Peijie |
| contents | We establish dimension-free quantum Talagrand-type inequalities with explicit constants on the quantum Boolean cube, via a unified variance-decay perspective. For individual observables, short-time variance decay along the depolarizing semigroup, with rates estimated through hypercontractivity, naturally yields Talagrand-type bounds. Within this framework, we derive Talagrand-type energy--variance and high-order influence--variance inequalities. From the former, we obtain quantum analogues of Talagrand's isoperimetric inequality, the Eldan--Gross inequality, and the Cordero-Erausquin--Eskenazis inequality; from the latter, we derive high-order quantum Talagrand--KKL-type and partial isoperimetric bounds. Altogether, our work provides a variance-decay framework of broad applicability, unifying first-order and high-order Talagrand-type phenomena. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01900 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum Talagrand-type Inequalities via Variance Decay Chang, Fan Li, Peijie Functional Analysis Combinatorics Probability 06E30, 46L53, 47D07 We establish dimension-free quantum Talagrand-type inequalities with explicit constants on the quantum Boolean cube, via a unified variance-decay perspective. For individual observables, short-time variance decay along the depolarizing semigroup, with rates estimated through hypercontractivity, naturally yields Talagrand-type bounds. Within this framework, we derive Talagrand-type energy--variance and high-order influence--variance inequalities. From the former, we obtain quantum analogues of Talagrand's isoperimetric inequality, the Eldan--Gross inequality, and the Cordero-Erausquin--Eskenazis inequality; from the latter, we derive high-order quantum Talagrand--KKL-type and partial isoperimetric bounds. Altogether, our work provides a variance-decay framework of broad applicability, unifying first-order and high-order Talagrand-type phenomena. |
| title | Quantum Talagrand-type Inequalities via Variance Decay |
| topic | Functional Analysis Combinatorics Probability 06E30, 46L53, 47D07 |
| url | https://arxiv.org/abs/2601.01900 |