Non-Associativity Induced Modifications of Open-System Quantum Dynamics: General Master Equation and a Two-Qubit Ising Case Study

TL;DR

This paper derives a modified quantum master equation incorporating weak nonassociativity effects, revealing how such deformations influence open-system dynamics, entanglement, and state purity in a two-qubit Ising model.

Contribution

It introduces a novel correction to the quantum master equation accounting for nonassociative operator products and demonstrates its impact on entanglement and coherence in a two-qubit system.

Findings

Nonassociativity suppresses steady-state entanglement by up to 59%.

It reduces the purity of quantum states and increases entropy.

The relaxation timescale remains unaffected by nonassociativity.

Abstract

Nonassociative deformations of phase-space structures arise naturally in the presence of magnetic charge, where the Jacobi identity for momentum components fails and the corresponding Moyal product becomes nonassociative. While such structures are well understood at the level of single-particle kinematics, their implications for open-system quantum dynamics remain largely unexplored. Here we derive a Born-Markov master equation for a system coupled to a bath when the underlying operator product is weakly nonassociative. The deformation enters through associators appearing in the second-order kernel, while pairwise operator products and dissipators retain their standard form. The resulting correction is dispersive and modifies the Liouville-von Neumann part of the generator without introducing additional dissipative channels. We then embed this structure into a two-qubit transverse-field Ising model using a Stratonovich-Weyl representation and an Ising-aligned twisted Poisson structure. In the zero-temperature limit, the nonassociative terms produce a nonlinear correction in which the instantaneous population imbalance of each qubit feeds back into the dynamics as a state-dependent longitudinal field. Numerical simulations in the weak-coupling regime, where the Born-Markov derivation is quantitatively controlled, show that increasing the nonassociativity parameter suppresses steady-state entanglement by up to 59%, reduces purity, and increases entropy, while leaving the relaxation timescale set by the dissipative rates unchanged. These results demonstrate that weak nonassociativity manifests as a coherent, population-dependent deformation of open-system dynamics rather than an additional dissipative mechanism.