Persistence of quantum triality relations in open <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -dimensional systems

We examine the complementarity among coherence (visibility), predictability, and entanglement for qubit and qutrit systems subjected to noisy quantum channels. Using the system-path entanglement framework, analytical expressions for all three quantities are derived for two-, three-, and <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>n</a:mi> </a:math> -slit interferometric setups. The study first establishes the validity of the triality relation in ideal conditions and then investigates its behavior under amplitude and phase damping. We find that amplitude damping redistributes coherence and population imbalance without violating complementarity, while phase damping reduces coherence but leaves predictability unchanged. We further extend the study to non-Markovian amplitude damping, where environmental memory effects lead to nonmonotonic evolution and revivals of coherence and entanglement. These results demonstrate that the complementarity relation remains preserved even in open quantum systems, highlighting its robustness against decoherence for both Markovian and non-Markovian open-system dynamics and providing a unified analytical understanding of noisy quantum interferometry in low-dimensional systems.