Quantum-classical correspondence in quantum channels

TL;DR

This paper explores the correspondence between quantum and classical channels using Koopman operators, revealing how classical phase space structures influence quantum channel spectra and their classical limits.

Contribution

It introduces classical Koopman channels as analogs to quantum channels, enabling detailed comparison of quantum-classical correspondence at the channel level.

Findings

Quantum and classical channels can be directly compared using spectra.

The quantum channel's dominant modes relate to stable classical phase space regions.

In chaotic regimes, the spectrum exhibits a ring structure described by random matrix theory.

Abstract

Quantum channels describe subsystem or open system evolution. Using the classical Koopman operator that evolves functions on phase space, 4 classical Koopman channels are identified that are analogs of the 4 possible quantum channels in a bipartite setting. Thus when the complete evolution has a quantum-classical correspondence the correspondence at the level of the subunitary channels can be studied. The channels, both classical and quantum can be interpreted as noisy single particle systems. Having parallel classical and quantum operators gives us new access to study fine details of these major limiting theories. Using a coupled kicked rotor as a generic example, we contrast and compare spectra of the quantum and classical channel. The largest nontrivial mode of the quantum channel is seen to be mostly determined by the stable parts of the classical phase space, even those that are surprisingly small in relation to the scale of an effective $\hbar$. In the case when the dynamics has a significant fraction of chaos the spectrum has a prominent annular density that is approximately described by the single-ring theorem of random matrix theory, and the ring shrinks in size when the classical limit is approached. However, the eigenvalues and modes that survive the classical limit seem to be either scarred by unstable manifolds or, if they exist, stable periodic orbits.