Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits
Gergely Bunth, József Pitrik, Tamás Titkos, Dániel Virosztek
TL;DR
The paper develops a strong Kantorovich duality theory for a linearized version of a non-quadratic quantum optimal transport problem where transport is implemented by quantum channels. By applying Fenchel-Rockafellar duality to a primal-dual pair with operator-inequality and marginal constraints, it obtains a precise dual formulation and shows strong duality, including a useful corollary for factorized costs. The authors then expose concrete, analytically tractable cases on quantum bits, deriving closed-form optimal couplings and transport costs for commuting qubits with a symmetric cost and for single-observable costs, and they obtain explicit expressions for associated quantum Wasserstein distances and their quadratic divergences. They also demonstrate a triangle inequality for the induced divergences in the commuting-qubit regime, highlighting both the theoretical robustness and the potential for precise quantitative OT-type analysis in quantum information tasks.
Abstract
We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, we use this information on optimal solutions to give an analytical proof of the triangle inequality for the induced quantum Wasserstein divergences.