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

Eric A Carlen1, Jan Maas2

  • 1Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019 USA.

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|November 23, 2020
PubMed
Summary

This study introduces unified dynamical optimal transport metrics for quantum systems, generalizing existing methods for Markov chains and quantum dynamics. The research provides new tools for analyzing quantum information and dynamics using non-commutative geometry.

Keywords:
Functional inequalitiesGradient flowLindblad equationNon-commutative optimal transport

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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.