Two-qudit topological phase evolution under dephasing

In this work, we study a bipartite system composed by a pair of entangled qudits coupled to an environment. Initially, we derive a master equation and show how the dynamics can be restricted to a "diagonal" sector that includes a maximally entangled state (MES). Next, we solve this equation for mixed qutrit pairs and analyze the $I$-concurrence $C(t)$ for the effective state, which is needed to compute the geometric phase when the initial state is pure. Unlike (locally operated) isolated systems, the coupled system leads to a nontrivial time-dependence, with $C(t)$ generally decaying to zero at asymptotic times. However, when the initial condition gets closer to a MES state, the effective concurrence is more protected against the effects of decoherence, signaling a transition to an effective two-qubit MES state at asymptotic times. This transition is also observed in the geometric phase evolution, computed in the kinematic approach. Finally, we explore the system-environment coupling parameter space and show the existence of a Weyl symmetry among the various physical quantities.