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arXiv:2512.06940

Optimal Transport of a Free Quantum Particle and its Shape Space Interpretation

Bernadette Lessel

quant-phmath-phmath.FASubmitted Dec 7, 2025• Announced Dec 9, 2025• 0 citations
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Abstract

A solution of the free Schrödinger equation is investigated by means of Optimal transport. The curve of probability measures $μ_t$ this solution defines is shown to be an absolutely continuous curve in the Wasserstein space $W_2(\mathbb{R}^3)$. The optimal transport map from $μ_t$ to $μ_s$, the cost for this transport (i.e. the Wasserstein distance) and the value of the Fisher information along $μ_t$ are being calculated. It is finally shown that this solution of the free Schrödinger equation can naturally be interpreted as a curve in so-called Shape space, which forgets any positioning in space but only describes properties of shapes. In Shape space, $μ_t$ continues to be a shortest path geodesic.

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26 pages

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Abstract We show that the Schrödinger equation is a lift of Newton's third law of motion on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potentialμ→F(μ) is the sum of the total classical potential energy (V, μ) of the extended system and its Fisher information . The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures.
This thesis offers a mathematical framework to treat quantum dynamics without reference to a background structure, but rather by means of the change of the shape of the state. For this, Wasserstein geometry is used. The so called Shape space, then, is defined as the quotient space of a Wasserstein space modulo the action of the isometry group of the background space. On Shape space we find a natural metric distance, the Shape distance, of which we investigate topological, metric and geodesic properties. Canonically mapped solutions of the Schrödinger equation turn out to be naturally nice curves in Shape space and some even constitue geodesics there. To also be able to speak about infinitesimal change of shapes, a definition for a tangent space at a point in Shape space is defined and applied. Finally a notion of differentiability for maps between Wasserstein spaces is proposed.
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 cost associated with transforming one quantum state to another, is expected to have implications in quantum state discrimination and quantum metrology. In this work, following the formalism introduced in [De Palma, G. and Trevisan, D. Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] to compute the optimal transport plan between two quantum states, we give a general formula for the Wasserstein distance of order 2 between any two one-mode Gaussian states. We discuss how the Wasserstein distance between classical Gaussian distributions and the quantum Wasserstein distance by De Palma and Trevisan for thermal states can be recovered from our general formula for Gaussian states. This opens the path to directly compare various known distance measures with the Wasserstein distance through their closed-form solutions.
We show that the Schroedinger equation is a lift of Newton's law of motion on the space of probability measures, where derivatives are taken w.r.t. the Wasserstein Riemannian metric. Here the potential is the sum of the total classical potential energy of the extended system and its Fisher information. The precise relation is established via a well known ('Madelung') transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures
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