Abstract

 

​This work presents a geometrically explicit representation of spin and polarization measurements in which the state of an individual particle is described by a continuous directional amplitude defined on the unit sphere. Measurement outcomes are modeled as local projection operations acting on these amplitudes, with statistical predictions obtained through averaging over preparation-dependent distributions. 

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Within this formulation, the correlation structure of entangled photon pairs is derived directly from phase-coherent amplitude overlaps, where phase information is encoded geometrically through relative orientation. The resulting expression reproduces the standard quantum correlation function 𝐸(𝛼,𝛽)=cos⁡(2(𝛼−𝛽)) and exactly saturates the Tsirelson bound, without requiring additional assumptions beyond local projection geometry and phase tracking. 

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The analysis provides a unified geometric description of Stern–Gerlach filtering, polarization, and bipartite correlations, highlighting how phase coherence at the amplitude level governs the formation of quantum statistics prior to probability assignment.