Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

Area of Science:

• Quantum Information Theory
• Non-commutative Geometry
• Mathematical Physics

Background:

• Optimal transport metrics are crucial for comparing probability distributions.
• Dirichlet forms provide a framework for analyzing Markovian dynamics.
• Existing transport metrics are limited to specific systems like Markov chains and Lindblad equations.

Purpose of the Study:

• To develop a unified framework for dynamical optimal transport metrics on finite-dimensional algebras.
• To generalize existing transport metrics to a broader class of quantum systems.
• To establish connections between optimal transport, non-commutative geometry, and spectral analysis.

Main Methods:

• Utilizing symmetric Dirichlet forms on finite-dimensional algebras.
• Developing a non-commutative differential calculus.
• Applying the framework to arbitrary skew-derivations.

Main Results:

• A unified framework for dynamical optimal transport metrics.
• Generalization of transport metrics for Markov chains, Lindblad equations, and Fermi Ornstein-Uhlenbeck semigroup.
• Derivation of non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

Conclusions:

• The developed framework offers a powerful tool for studying quantum dynamics and information.
• The results provide new insights into the interplay between optimal transport and non-commutative geometry.
• This work paves the way for further research in quantum information theory and mathematical physics.