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arXiv · 1811.04572

Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems

Eric A. Carlen·Jan Maas
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Abstract

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $C^*$-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

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Eric A. Carlen, Jan Maas. 2018-11-12. Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems. https://doi.org/10.1007/s10955-019-02434-w

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