Matt Hoogsteder-Riera, John Calsamiglia, and Andreas Winter have calculated optimal quantum transport analytically by generalizing a classical cost theory. The researchers built on Monge’s classical work, replacing probability densities with and transport plans with “completely positive and trace preserving maps”. This work introduces a mathematical object called a “stote”, a combination of quantum state and process, to integrate these elements. The team demonstrates that this quantum transport cost differs from its classical counterpart, suggesting a fundamental departure in understanding how resources are optimally distributed in quantum systems.

Quantum Transport Costs Defined via “Stotes”

The analytical calculation of quantum transport costs relies on a newly defined mathematical object called a “stote,” which integrates a quantum state with the process acting upon it. This “stote” functions analogously to joint probability distributions in classical theory, providing a unified description for calculations. Researchers postulate that the quantum transport cost is a linear function of this stote, allowing for exploration of its properties and the determination of the optimal transport cost between quantum states. Following this framework, the team analyzed the case of unitary invariant cost, achieving analytical solutions for many calculations.

These results indicate a qualitative difference between their quantum transport cost and the classical Monge transport, a long-standing mathematical problem concerning optimal distribution of resources. The classical problem asks how to build fortifications with minimal material displacement, or how to optimally distribute resources from production to distribution centers. This work extends that framework to quantum systems. “We seek a quantum formulation that mirrors the structure of the classical theory as closely as possible,” the researchers write, emphasizing their goal of building upon established principles.

“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,” the paper details, outlining the core mathematical construction. The researchers define a family of quantum transport costs and study the optimization problem between quantum states, investigating the mathematical properties of the resulting framework. This work lays the groundwork for future studies aimed at identifying physically meaningful notions of quantum transport.

Unitary Invariance Enables Analytical Calculations

Analytical solutions for quantum transport costs emerge from a focus on unitary invariance, allowing for detailed comparisons with classical counterparts. The team demonstrated that, under these conditions, the quantum transport cost diverges qualitatively from the classical Monge transport cost, suggesting fundamental differences in how information moves in these systems. This construct integrates the density matrix, representing the quantum state, with the completely positive and trace preserving map, which defines the transport plan.

The researchers used the properties of the Jordan product to define this integration, facilitating the analytical treatment of the transport problem. Further analysis revealed that the quantum transport cost exhibits unique characteristics when restricted to unitary invariant scenarios.

Unitary invariance, a symmetry condition, allows for the identification of specific cost functions that can be solved without resorting to approximations. The team’s findings indicate that the resulting quantum cost is a fundamentally different entity than the classical cost. This divergence suggests that quantum systems may possess inherent transport properties not found in their classical counterparts, potentially opening new avenues for quantum information processing and control.

The mathematical tools employed in this study build upon established concepts from convex optimization and functional analysis. The researchers used techniques for analyzing convex cones and semidefinite programming to characterize the properties of the quantum transport cost. These methods, combined with the novel “stote” formalism, provide a powerful toolkit for investigating optimal transport in quantum systems. The work also draws on earlier research into the Monge-Kantorovich problem, adapting its principles to the quantum realm.

This connection to classical optimal transport theory allows for a rigorous comparison of the two frameworks, highlighting both similarities and differences.

Classical Transport Theory and Quantum Generalization

The analytical results demonstrate a qualitative divergence between quantum and classical transport costs. This departure from classical models is revealed through calculations of a metric used to assess the efficiency of transporting quantum states. The team’s work builds upon the Monge transport problem, a classical framework for minimizing the cost of moving mass distributions, and extends it to the quantum realm. The researchers note that their findings have implications extending beyond theoretical mathematics.

Foundations of Completely Positive Trace-Preserving Maps

Analytical solutions for unitary invariant cost reveal a departure from classical transport behavior, demonstrating that the quantum transport cost is not simply a quantum analogue of its classical counterpart. The analytical tractability of the unitary invariant cost case enabled the researchers to derive concrete results and validate their theoretical framework.

Further analysis revealed that the framework extends beyond simply mirroring classical transport; it introduces qualitatively different behaviors. The researchers’ work approaches the problem of constructing a quantum analogue of the classical transport cost theory of Monge.

👉 More information
🗞 Approach to optimal quantum transport via states over time
✍️ Matt Hoogsteder-Riera, John Calsamiglia and Andreas Winter
🧠 DOI: https://quantum-journal.org/papers/q-2026-10-14-2236/

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