A universal complementarity identity for polarized double-slit interferometry

TL;DR

This paper derives a universal identity linking key measurable quantities in polarized double-slit interferometry, unifying existing relations and providing a new framework for understanding coherence and mixedness.

Contribution

It establishes an exact, universal complementarity identity among four invariants in polarized double-slit experiments, unifying and extending prior relations.

Findings

The identity V_A^2 + V_N^2 + P^2 + I^2 = 1 holds universally for normalized path-polarization states.

Separating V into V_A and V_N reveals the antisymmetric coherence sector as key to phase-sensitive information.

The invariants parametrize the minimal exponential family, with I^2 representing residual mixedness.

Abstract

We establish an exact identity among four dimensionless invariants accessible by standard polarimetric and interferometric measurements in a polarized double-slit experiment: the in-phase and quadrature components V_A and V_N of fringe visibility, the path predictability P, and the mixedness I of the path-reduced state satisfy V_A^2 + V_N^2 + P^2 + I^2 = 1. The identity is a universal algebraic consequence of the positivity of the reduced state and holds for every normalized path-polarization density matrix. It unifies the Englert-Greenberger-Yasin and Jakob-Bergou relations, separates the two operationally distinct components of visibility measurable by phase-shifted interferometry, and admits a natural interpretation within the Jaynes maximum-entropy framework: the three path invariants parametrize the minimal exponential family on the accessible algebra, while I^2 emerges as the residual mixedness that saturates the positivity bound. The separation V^2 = V_A^2 + V_N^2 identifies the antisymmetric sector of the coherence matrix rho = A + iN as the specific substrate of phase-sensitive information and permits a sector-resolved diagnosis of environmental coupling.