A Generalized Density Matrix Formalism for N-State Quantum Systems with Deterministic Phase Shift Interactions

This research paper presents a mathematical framework that models how quantum systems interact and lose coherence. While standard quantum mechanics often treats environmental interference and phase shifts as completely random noise, this paper develops a deterministic model to track these phase changes precisely. By defining these interactions analytically, the framework creates a structured protocol to predict and calibrate phase errors. To prove its practical value, the model was tested on real quantum computing hardware (IBM Q), demonstrating that mapping these deterministic shifts can actively clean up systematic errors and improve gate fidelity. Ultimately, the paper provides a fresh perspective on the boundary between quantum superposition and classical measurement by showing that certain "random" decoherence effects can actually be tracked and calculated. "Note: This theoretical framework was originally formulated and completed on March 15, 2025."