Statistical Mechanics of Stochastic Quantum Control: $d$-adic R\'enyi Circuits

TL;DR

This paper explores the connection between classical and quantum models of stochastic control in many-body systems, revealing phase transitions in chaos, control, and entanglement through effective statistical mechanics models.

Contribution

It introduces a unified framework linking classical $d$-adic Rényi maps, quantum analogs, and Potts models to analyze phase transitions in quantum control and entanglement.

Findings

Entanglement transition belongs to bond-percolation universality class.

Control transition is governed by a classical random walk.

Transitions merge as model parameters vary.

Abstract

The dynamics of quantum information in many-body systems with large onsite Hilbert space dimension admits an enlightening description in terms of effective statistical mechanics models. Motivated by this fact, we reveal a connection between three separate models: the classically chaotic $d$-adic R\'{e}nyi map with stochastic control, a quantum analog of this map for qudits, and a Potts model on a random graph. The classical model and its quantum analog share a transition between chaotic and controlled phases, driven by a randomly applied control map that attempts to order the system. In the quantum model, the control map necessitates measurements that concurrently drive a phase transition in the entanglement content of the late-time steady state. To explore the interplay of the control and entanglement transitions, we derive an effective Potts model from the quantum model and use it to probe information-theoretic quantities that witness both transitions. The entanglement transition is found to be in the bond-percolation universality class, consistent with other measurement-induced phase transitions, while the control transition is governed by a classical random walk. These two phase transitions merge as a function of model parameters, consistent with behavior observed in previous small-size numerical studies of the quantum model.