Measurement-Induced Boolean Dynamics for Open Quantum Networks

TL;DR

This paper explores how sequential measurements in open quantum networks induce probabilistic Boolean dynamics, revealing structural properties and their relation to the quantum system's relaxation behavior and interaction graph.

Contribution

It provides an explicit representation of Boolean network transitions from quantum master equations and links network properties to quantum relaxation and graph Laplacians.

Findings

Relaxing quantum dynamics lead to irreducible, aperiodic Boolean chains.

Measurement outcomes form probabilistic Boolean networks with explicit transition representations.

Quantum Laplacian encodes communication classes in non-relaxing quantum networks.

Abstract

In this paper, we study the recursion of measurement outcomes for open quantum networks under sequential measurements. Open quantum networks are networked quantum subsystems (e.g., qubits) with the state evolutions described by a continuous Lindblad master equation. When measurements are performed sequentially along such continuous dynamics, the quantum network states undergo random jumps and the corresponding measurement outcomes can be described by a vector of probabilistic Boolean variables. The induced recursion of the Boolean vectors forms a probabilistic Boolean network. First of all, we show that the state transition of the induced Boolean networks can be explicitly represented through realification of the master equation. Next, when the open quantum dynamics is relaxing in the sense that it possesses a unique equilibrium as a global attractor, structural properties including absorbing states, reducibility, and periodicity for the induced Boolean network are direct consequences of the relaxing property. Particularly, we show that generically, relaxing quantum dynamics leads to irreducible and aperiodic chains for the measurement outcomes. Finally, we show that for quantum consensus networks as a type of non-relaxing open quantum network dynamics, the communication classes of the measurement-induced Boolean networks are encoded in the quantum Laplacian of the underlying interaction graph.