Spectral transitions and universal steady states in random Kraus maps and circuits

TL;DR

This paper investigates the spectral and steady-state properties of random Kraus maps in open quantum systems, revealing universal behaviors and phase transitions that are consistent across different models and system sizes.

Contribution

It introduces a comprehensive analysis of spectral transitions and steady states in random Kraus maps using RMT, demonstrating their universality and connection to Lindbladian dynamics.

Findings

Spectral transition from disk to annulus in the complex plane at critical dissipation.

Steady state remains unaffected by spectral transition, showing universality.

Local Kraus circuits replicate the properties of nonlocal Kraus maps, indicating robustness.

Abstract

The study of dissipation and decoherence in generic open quantum systems recently led to the investigation of spectral and steady-state properties of random Lindbladian dynamics. A natural question is then how realistic and universal those properties are. Here, we address these issues by considering a different description of dissipative quantum systems, namely, the discrete-time Kraus map representation of completely positive quantum dynamics. Through random matrix theory (RMT) techniques and numerical exact diagonalization, we study random Kraus maps, allowing for a varying dissipation strength, and their local circuit counterpart. We find the spectrum of the random Kraus map to be either an annulus or a disk inside the unit circle in the complex plane, with a transition between the two cases taking place at a critical value of dissipation strength. The eigenvalue distribution and the spectral transition are well described by a simplified RMT model that we can solve exactly in the thermodynamic limit, by means of non-Hermitian RMT and quaternionic free probability. The steady state, on the contrary, is not affected by the spectral transition. It has, however, a perturbative crossover regime at small dissipation, inside which the steady state is characterized by uncorrelated eigenvalues. At large dissipation (or for any dissipation for a large-enough system), the steady state is well described by a random Wishart matrix. The steady-state properties thus coincide with those already observed for random Lindbladian dynamics, indicating their universality. Quite remarkably, the statistical properties of the local Kraus circuit are qualitatively the same as those of the nonlocal Kraus map, indicating that the latter, which is more tractable, already captures the realistic and universal physical properties of generic open quantum systems.