Related papers: Quantum optimal transport is cheaper

Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal…

We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincar\'e, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying…

Controlling the $\mathcal W_\infty$ Wasserstein distance by the $\mathcal W_p$ Wasserstein distance is interesting both for theorical and numerical applications. A first paper on this problem was written several years ago [3]. Some year…

We propose a new generalization to quantum states of the Wasserstein distance, which is a fundamental distance between probability distributions given by the minimization of a transport cost. Our proposal is the first where the transport…

We analyze a quantum version of the Monge--Kantorovich optimal transport problem. The quantum transport cost related to a Hermitian cost matrix $C$ is minimized over the set of all bipartite coupling states $\rho^{AB}$ with fixed reduced…

We resolve a conjecture of De Palma and Trevisan by proving the triangle inequality for a quantum 2-Wasserstein distance. The proof relies on complex analysis methods to establish a new integral representation of the cost in the optimal…

We define the quantum Wasserstein distance such that the optimization of the coupling is carried out over bipartite separable states rather than bipartite quantum states in general, and examine its properties. Surprisingly, we find that the…

The theory of optimal transport of probability measures has wide-ranging applications across a number of different fields, including concentration of measure, machine learning, Markov chains, and economics. The generalisation of optimal…

We show that the quantum generalization of the $2$-Wasserstein distance proposed by Chakrabarti et al. is not monotone under partial traces. This disproves a recent conjecture by Friedland et al. Finally, we propose a stabilized version of…

This text is a set of lecture notes for a 4.5-hour course given at the Erd\"os Center (R\'enyi Institute, Budapest) during the Summer School "Optimal Transport on Quantum Structures" (September 19th-23rd, 2023). Lecture I introduces the…

We introduce folded optimal transport, as a method to extend a cost or distance defined on the extreme boundary of a convex to the whole convex, related to convex extension. This construction broadens the framework of standard optimal…

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…

A recent development of the studies on classical and quasi-classical properties of supersymmetric quantum mechanics in Witten's version is reviewed. First, classical mechanics of a supersymmetric system is considered. Solutions of the…

We approach the problem of constructing a quantum analogue of the immensely fruitful classical transport cost theory of Monge from a new angle. Going back to the original motivations, by which the transport is a bilinear function of a mass…

We consider several definitions of the quantum Wasserstein distance based on an optimization over general bipartite quantum states with given marginals. Then, we examine the quantities obtained after the optimization is carried out over…

A method to optimize the cost of a quantum channel is developed. The goal is to determine the cheapest channel that produces prescribed output states for a given set of input states. This is essentially a quantum version of optimal…

Classical and quantum physics represent two distinct theories; however, quantum physics is regarded as the more fundamental of the two. It is posited that classical mechanics should arise from quantum mechanics under certain limiting…

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being "classical" or "quantum" ceases to be a basic conceptual difference. The dynamics differs,…

We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant…

The guesswork of a classical-quantum channel quantifies the cost incurred in guessing the state transmitted by the channel when only one state can be queried at a time, maximized over any classical pre-processing and minimized over any…