[Paper Review] On the Matrix Monge-Kantorovich Problem

Yongxin Chen, Wilfrid Gangbo+2 more|arXiv (Cornell University)|Jan 11, 2017

Geometric Analysis and Curvature Flows6 references9 citations

TL;DR

This paper formulates a matrix-valued counterpart of the classical Monge-Kantorovich optimal transport problem for positive definite density matrices, establishing a non-commutative Wasserstein metric via convex optimization. It proves strong duality, a Poincaré-Wirtinger-type inequality, and a Lax-Hopf-Oleinik-type result, generalizing the Benamou-Brenier dynamic formulation to quantum density matrices with applications to quantum information and transport theory.

ABSTRACT

The classical Monge-Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the "Wasserstein distance." In this note, we formulate a natural matrix counterpart of the MK problem for positive definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimization problem, strong duality, an analogue of the Poincare-Wirtinger inequality, and a Lax-Hopf-Oleinik type result.

Motivation & Objective

• To extend optimal transport theory to the non-commutative setting of positive definite density matrices.
• To formulate a dynamic, convex optimization-based Wasserstein metric for quantum states analogous to the classical Benamou-Brenier approach.
• To establish strong duality and duality-based characterizations in the matrix setting.
• To derive geometric and analytic properties such as a Poincaré-Wirtinger-type inequality and Hamiltonian conservation.
• To explore implications for quantum channels and quantum information geometry.

Proposed method

• Formulates a dynamic optimal transport problem on the space of positive definite density matrices using a continuity equation derived from the Lindblad equation.
• Defines a kinetic energy functional based on the matrix gradient operator ∇L, leading to a Riemannian structure on the space of density matrices.
• Applies convex optimization techniques to the action integral, ensuring the existence of minimizers and establishing strong duality.
• Uses duality to derive a Kantorovich-type dual formulation involving a dual variable λ satisfying a Hamilton-Jacobi inequality.
• Implements a time-reparametrization argument using diffeomorphisms to prove conservation of the Hamiltonian and time-homogeneity of the metric.
• Relies on the structure of the space of Hermitian matrices and the trace inner product to define the geometry and inner products in the matrix setting.

Experimental results

Research questions

• RQ1 Can the classical Monge-Kantorovich optimal transport framework be generalized to the non-commutative setting of density matrices in quantum mechanics?
• RQ2 Does a dynamic, convex formulation of optimal transport exist for quantum states that preserves the Wasserstein metric structure?
• RQ3 What is the dual formulation of the matrix optimal transport problem, and how does it relate to Kantorovich duality?
• RQ4 How do geometric and analytic properties such as the Poincaré-Wirtinger inequality and Hamiltonian conservation generalize in the matrix setting?
• RQ5 Can the dynamic Benamou-Brenier approach be adapted to define a quantum Wasserstein metric that is more versatile than the classical MK formulation?

Key findings

• The matrix Monge-Kantorovich problem is formulated as a convex optimization problem with strong duality, enabling numerical implementation.
• A Poincaré-Wirtinger-type inequality is established in the non-commutative setting, providing a spectral bound on transport cost.
• The Hamiltonian F(ρ(t), m(t)) is conserved along optimal paths, implying constant kinetic energy in the dynamic formulation.
• The Wasserstein distance satisfies a time-homogeneous scaling: W₂(ρ(s), ρ(t)) = (t−s)W₂(ρ₀, ρ₁), showing linear growth.
• The dual formulation yields a characterization of the transport cost via a subsolution condition on λ, with equality in the Hamilton-Jacobi inequality along optimal paths.
• The results extend to infinite-dimensional settings, suggesting broader applicability in quantum information and functional analysis.