Stability of quantum chaos against weak non-unitarity

TL;DR

This paper investigates how weak non-unitarity affects quantum chaos, showing that spectral properties of the evolution operator determine the delay in state purification and relate to chaos signatures.

Contribution

It reveals that eigenvalue distributions with sharp edges in the complex plane cause slow purification, linking spectral features to quantum chaos resilience under non-unitarity.

Findings

Eigenvalue rings with sharp edges cause exponential delay in purification.

Spectral form factor indicates level repulsion and chaos signatures.

Eigenvalue distribution shape influences sensitivity to initial conditions.

Abstract

We study the quantum dynamics generated by the repeated action of a non-unitary evolution operator on a system of qubits. Breaking unitarity can lead to the purification of mixed initial states, which corresponds to the loss of sensitivity to initial conditions, and hence the absence of a key signature of dynamical chaos. However, the scrambling of quantum information can delay purification to times that are exponential in system size. Here we study purification in systems whose evolution operators are fixed in time, where all aspects of the dynamics are in principle encoded in spectral properties of the evolution operator for a single time step. The operators that we study consist of global Haar random unitary operators and non-unitary single-qubit operations. We show that exponentially slow purification arises from a distribution of eigenvalues in the complex plane that forms a ring with sharp edges at large radii, with the eigenvalue density exponentially large near these edges. We argue that the sharp edges of the eigenvalue distribution arise from level attraction along the radial direction in the complex plane. By calculating the spectral form factor we also show that there is level repulsion around the azimuthal direction, even close to the outer edge of the ring of eigenvalues. Our results connect this spectral signature of quantum chaos to the sensitivity of the system to its initial conditions.