A Trotter-Kato Theorem for Quantum Markov Limits

TL;DR

This paper proves the convergence of certain quantum dynamics to a quantum stochastic evolution using the Trotter-Kato theorem, establishing a link between singular Hamiltonians and quantum Ito evolutions.

Contribution

It extends the Trotter-Kato theorem to quantum systems, demonstrating convergence of Hamiltonian dynamics to quantum stochastic evolutions in the singular limit.

Findings

Convergence of unitary dynamics to Hudson-Parthasarathy type evolution.

Graph limit convergence of Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Hamiltonian.

Establishment of a rigorous link between singular Hamiltonians and quantum stochastic processes.

Abstract

Using the Trotter-Kato theorem we prove the convergence of the unitary dynamics generated by an increasingly singular Hamiltonian in the case of a single field coupling. The limit dynamics is a quantum stochastic evolution of Hudson-Parthasarathy type, and we establish in the process a graph limit convergence of the pre-limit Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Hamiltonian generating the quantum Ito evolution.