A Dual Formula for the Noncommutative Transport Distance

Area of Science:

• Mathematical Physics
• Optimal Transport Theory
• Noncommutative Geometry

Background:

• The Carlen-Maas noncommutative transport distance provides a framework for measuring distances in noncommutative spaces.
• The Becker-Li entropic regularization offers a smoothed version of this distance, crucial for computational and analytical tractability.
• Wasserstein distances are fundamental in optimal transport, with dual formulations offering powerful analytical tools.

Purpose of the Study:

• To investigate the properties of the noncommutative transport distance and its entropic regularization.
• To establish a novel duality formula for the entropic regularization of the noncommutative transport distance.
• To connect this new formula to existing concepts in optimal transport and Hamilton-Jacobi-Bellmann equations.

Main Methods:

• Utilizing techniques from optimal transport theory.
• Applying concepts from functional analysis and partial differential equations.
• Developing a novel duality approach based on subsolutions of Hamilton-Jacobi-Bellmann equations.

Main Results:

• A new duality formula for the entropic regularization of the noncommutative transport distance is derived.
• This formula is shown to be a quantum analogue of the dual Benamou-Brenier formulation of the Wasserstein distance.
• The formula is expressed in terms of subsolutions to a relevant Hamilton-Jacobi-Bellmann equation.

Conclusions:

• The derived duality formula offers a new perspective on noncommutative optimal transport.
• This work bridges concepts from noncommutative geometry, optimal transport, and viscosity solutions.
• The findings have potential implications for quantum information theory and mathematical physics.