Ergodic dynamics in iterated quantum protocols

TL;DR

This paper investigates measurement-induced nonlinear dynamics in iterated quantum protocols, revealing ergodic behavior, robustness to noise, and the coexistence of mixing and purification, with implications for quantum chaos and information processing.

Contribution

It introduces a family of ergodic-like quantum protocols, analyzes their robustness under noise, and uncovers the coexistence of mixing and purification phenomena in quantum dynamics.

Findings

Global chaos achieved with specific unitaries exploring the Bloch sphere

Maximally mixed state acts as an attractor for noisy inputs

Protocols exhibit quasi-ergodicity with rapid state spreading and purity decay

Abstract

We study measurement-induced nonlinear dynamics generated by an iterated quantum protocol combining an entangling gate, a single-qubit rotation, and post-selection. For pure single-qubit inputs, a particular choice of the single-qubit unitary yields globally chaotic, strongly mixing dynamics that explores the entire Bloch sphere, providing a physical realization of ergodic behavior in a complex map. We extend the analysis to realistic, noisy preparation by considering mixed initial states and the induced nonlinear evolution inside the Bloch sphere. Numerical results show that the maximally mixed state is an attractor for mixed inputs, although many trajectories exhibit transient increases in purity before ultimately converging. To quantify robustness against noise, we introduce a practical notion of quasi-ergodicity: ensembles prepared in a small angular patch at fixed purity rapidly spread to cover all directions, while the purity gradually decreases toward its minimal value. By varying the final single-qubit gate, we identify a broad family of protocols that remain ergodic-like for pure states, supported by consistent diagnostics including the absence of attracting cycles, agreement of time and ensemble statistics, rapid spreading from localized regions, and exponential sensitivity to initial conditions. Away from the special globally mixing case, the mixed-state dynamics can change qualitatively: for most ergodic-like parameters, a finite subset of noisy inputs is driven toward purification rather than complete mixing, demonstrating the coexistence of statistical mixing and purification within a single iterated protocol.