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Abstract 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.
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Graph Hörmander Systems
Abstract This paper extends the Bakry-Émery criterion relating the Ricci curvature and logarithmic Sobolev inequalities to the noncommutative setting. We obtain easily computable complete modified logarithmic Sobolev inequalities of graph Laplacians and Lindblad operators of the corresponding graph Hörmander systems. We develop the anti-transference principle stating that the matrix-valued modified logarithmic Sobolev inequalities of sub-Laplacian operators on a compact Lie group are equivalent to such inequalities of a family of the transferred Lindblad operators with a uniform lower bound.
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- PAR ID:
- 10533844
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Annales Henri Poincaré
- Volume:
- 26
- Issue:
- 8
- ISSN:
- 1424-0637
- Format(s):
- Medium: X Size: p. 2683-2736
- Size(s):
- p. 2683-2736
- Sponsoring Org:
- National Science Foundation
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