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This paper advances theoretical quantum computing by reformulating classical optimal transport theory into a quantum framework using density matrices and completely positive maps. By introducing the "state over time" concept, the authors derive analytically solvable costs that reveal qualitative differences from their classical Monge counterparts. These findings provide foundational mathematical tools for optimizing state manipulation in future quantum processors.
Approach to optimal quantum transport via states over time
arXiv:2504.04856v3 Announce Type: replace Abstract: 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 distribution (without loss of generality a probability density) and a transport plan (a stochastic kernel), we explore the quantum version where the mass distribution is generalised to a density matrix, and the transport plan to a completely positive and trace preserving map. % These two data are naturally integrated into their Jordan product, which is called state over time (``stote''), and the transport cost is postulated to be a linear function of it. We explore the properties of this transport cost, as well as the optimal transport cost between two given states (simply the minimum cost over all suitable transport plans). After that, we analyse in considerable detail the case of unitary invariant cost, for which we can calculate many costs analytically. These findings suggest that our quantum transport cost is qualitatively different from Monge's classical transport.
Abstract: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 distribution (without loss of generality a probability density) and a transport plan (a stochastic kernel), we explore the quantum version where the mass distribution is generalised to a density matrix, and the transport plan to a completely positive and trace preserving map. % These two data are naturally integrated into their Jordan product, which is called state over time (``stote''), and the transport cost is postulated to be a linear function of it. We explore the properties of this transport cost, as well as the optimal transport cost between two given states (simply the minimum cost over all suitable transport plans). After that, we analyse in considerable detail the case of unitary invariant cost, for which we can calculate many costs analytically. These findings suggest that our quantum transport cost is qualitatively different from Monge's classical transport.
arXiv:2504.04856 [quant-ph]
(or arXiv:2504.04856v3 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2504.04856 Focus to learn more arXiv-issued DOI via DataCite
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