Stochastic Krylov Dynamics: Revisiting Operator Growth in Open Quantum Systems

TL;DR

This paper extends the geometric understanding of operator growth in quantum systems to open systems, showing environmental effects induce stochasticity and alter complexity growth.

Contribution

It introduces a Schwinger-Keldysh framework to derive an effective action for operator growth under Lindblad dynamics, revealing non-Hamiltonian, stochastic phase-space behavior.

Findings

Environmental coupling induces diffusion in Krylov space.

Exponential operator growth is suppressed beyond a certain scale.

Dissipation acts as a relevant perturbation to chaotic operator dynamics.

Abstract

In closed quantum systems, Krylov complexity admits a geometric description; operator growth is equivalent to Hamiltonian flow in an emergent phase space whose structure is fixed by the Lanczos coefficients. We show that this picture survives, albeit in a fundamentally altered form, once the system is coupled to an environment.Using a Schwinger-Keldysh formulation of the full counting statistics of the Krylov position, we derive an effective action for operator growth under Lindblad dynamics. Even for the minimal case of dephasing, the phase-space dynamics ceases to be Hamiltonian; environmental coupling generates diffusion in the variable conjugate to Krylov depth, converting deterministic trajectories in to stochastic ones. The hyperbolic mechanism underlying exponential complexity growth is therefore broadened and, beyond a parametrically controlled scale, destroyed.This identifies dissipation as a relevant perturbation of the chaotic Krylov fixed point and reveals operator growth in open systems as a problem of stochastic dynamics in an emergent phase space.