  This article introduces a new notion of optimal transport (OT) between tensor
fields, which are measures whose values are positive semidefinite (PSD)
matrices. This "quantum" formulation of OT (Q-OT) corresponds to a relaxed
version of the classical Kantorovich transport problem, where the fidelity
between the input PSD-valued measures is captured using the geometry of the
Von-Neumann quantum entropy. We propose a quantum-entropic regularization of
the resulting convex optimization problem, which can be solved efficiently
using an iterative scaling algorithm. This method is a generalization of the
celebrated Sinkhorn algorithm to the quantum setting of PSD matrices. We extend
this formulation and the quantum Sinkhorn algorithm to compute barycenters
within a collection of input tensor fields. We illustrate the usefulness of the
proposed approach on applications to procedural noise generation, anisotropic
meshing, diffusion tensor imaging and spectral texture synthesis.
