Quantum Chaos and Quantum Optimal Transport

Defining chaos in quantum mechanics is notoriously difficult because quantum states are extended distributions in a non-commutative phase space, unlike the point-like trajectories of classical chaos. This paper asks whether the theory of quantum optimal transport—which measures the cost of rearranging quantum distributions—can provide a rigorous, geometric foundation for defining quantum Lyapunov exponents that are both well-behaved and physically meaningful.

To bridge the gap between classical and quantum chaos, the authors leverage quantum optimal transport theory. They define a quantum Lyapunov exponent based on the exponential growth of the optimal transport distance between evolved quantum states. By establishing a set of natural axioms for these distances, they prove that their quantum Lyapunov exponent is finite and recovers the classical global expansion coefficient in the semiclassical limit. They further compare this metric-based approach to the growth of out-of-time-order correlators (OTOCs), which probe chaos through operator commutators.

The authors demonstrate that quantum optimal transport distances are well-approximated by classical Wasserstein distances in the semiclassical limit, up to corrections at the scale of ℏ. Their defined quantum Lyapunov exponent is shown to equal the classical global expansion coefficient, and in many standard settings, it coincides with the maximal classical Lyapunov exponent. Furthermore, they show that while OTOCs provide a state-weighted second-moment probe of instability, the quantum optimal transport exponent captures the optimized metric expansion of quantum states. This provides a unified mathematical structure that connects few-body and many-body quantum dynamics to classical chaotic theory.

This work provides a missing geometric link in quantum chaology. By moving away from ad-hoc diagnostics and toward an axiomatic, transport-based framework, researchers can now rigorously quantify chaotic behavior in quantum systems. This approach not only clarifies the interpretation of OTOCs but also opens the door to adapting other classical concepts—such as attractors and structural stability—to the quantum domain.