Cohesion accounting of complementarity in path--polarization interferometry

Two-path complementarity in polarized interferometric fields is reconsidered by retaining the complete path–polarization density matrix instead of reducing the description to the path degree of freedom from the outset. The familiar relation connecting the Cartesian visibility components, path predictability, and reduced-state mixedness is recovered as a marginal consequence of the reduced path state and is not interpreted as a new complementarity law. Attention is focused instead on the full path–polarization description in a real reference basis adapted to the path and linear-polarization degrees of freedom. Within this framework, the normalized purity separates naturally into path, polarization, and path–polarization-correlation contributions, while the antisymmetric sector provides a sector-resolved measure of cohesion. The resulting decomposition identifies which parts of the complete state store phase-sensitive interferometric coherence and which contributions are removed when polarization is traced out. The formalism therefore provides a sector-resolved accounting of complementarity within the full path–polarization state and clarifies the connection between reduced visibility loss, polarization marking, path–polarization correlations, and quantum-eraser recovery. The present article establishes the framework and associated purity decomposition; a more detailed exploration of full path–polarization complementarity and its dynamical aspects is left for future work.