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In this work, we study a bipartite system composed by a pair of entangledqudits coupled to an environment. Initially, we derive a master equation andshow how the dynamics can be restricted to a "diagonal" sector that includes amaximally entangled state (MES). Next, we solve this equation for mixed qutritpairs and analyze the $I$-concurrence $C(t)$ for the effective state, which isneeded to compute the geometric phase when the initial state is pure. Unlike(locally operated) isolated systems, the coupled system leads to a nontrivialtime-dependence, with $C(t)$ generally decaying to zero at asymptotic times.However, when the initial condition gets closer to a MES state, the effectiveconcurrence is more protected against the effects of decoherence, signaling atransition to an effective two-qubit MES state at asymptotic times. Thistransition is also observed in the geometric phase evolution, computed in thekinematic approach. Finally, we explore the system-environment couplingparameter space and show the existence of a Weyl symmetry among the variousphysical quantities.

Two-qudit topological phase evolution under dephasing