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arXiv:2606.12045

A Geometric Family of Correlations Containing the Quantum Singlet

E. Aldo Arroyo

quant-phSubmitted Jun 10, 2026• Announced Jun 11, 2026• 0 citations
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Abstract

We introduce a geometrically constrained hidden-variable framework that generates a family of correlations parametrized by a boundary function, within which the quantum singlet correlation appears as a particular member. Exact expressions for the correlation function are derived. Several structural results are established, including admissibility conditions, symmetry properties, a universal stationary point of the associated CHSH function, and an exact relation between the CHSH value at $ν=π/4$ and a geometric contrast measure defined on the underlying hidden-variable distributions. Rather than treating the quantum singlet correlation as an isolated target to be reproduced, the present framework places it within a broader geometric structure of correlations. These results suggest the existence of a nontrivial geometric structure underlying the family of correlations and motivate the search for a principle capable of selecting the quantum singlet solution from within that family.

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A Geometric Approach to Strongly Correlated Bosons: From NNN-Representability to the Generalized BEC Force
Building on recent advances in reduced density matrix theory, we develop a geometric framework for describing strongly correlated lattice bosons. We first establish that translational symmetry, together with a fixed pair interaction, enables an exact functional formulation expressed solely in terms of momentum occupation numbers. Employing the constrained-search formalism and exploiting a geometric correspondence between NNN-boson configuration states and their one-particle reduced density matrices, we derive the general form of the ground-state functional. Its structure highlights the omnipresent significance of one-body NNN-representability: (i) the domain is exactly determined by the NNN-representability conditions; (ii) at its boundary, the gradient of the functional diverges repulsively, thereby generalizing the recently discovered Bose-Einstein condensate (BEC) force; and (iii) an explicit expression for this boundary force follows directly from geometric arguments. These key results are demonstrated analytically for few-site lattice systems, and we illustrate the broader significance of our functional form in defining a systematic hierarchy of functional approximations.
A local hidden variable model exploiting the detection loophole to reproduce exactly the quantum correlation of the singlet state is presented. The model is shown to be compatible with both the CHSH and the CH Bell inequalities. Moreover, it bears the same rotational symmetry as spins. The reason why the model can reproduce the quantum correlation without violating the Bell theorem is that in the model the efficiency of the detectors depends on the local hidden variable. On average the detector efficiency is limited to 75%.
Bell’s inequality relies on a structural feature of the standard Bell-local model: the potential outcomes for all relevant measurement settings are jointly represented on a single hidden-variable space. This joint representation is not enforced by locality alone, and its status becomes nontrivial once the physical realization of measurement events and their associated symmetry structures are taken into account. We develop a symmetry-sensitive framework in which each measurement event is associated with an event-dependent symmetry action on the hidden-variable space. Within this framework, symmetry-compatible joint representability requires that these actions be compatible with a single equivariant hidden-variable model. We show that, for the symmetry structures considered here, no such common realization exists. Consequently, the standard jointly represented CHSH construction cannot be carried out within this symmetry-restricted class. The obstruction is algebraic rather than causal: spacelike separation rules out direct causal influence between the measurement events, but does not ensure compatibility of their symmetry actions. The analysis brings into view a representational structure beneath the standard Bell construction: joint representability is not only a probabilistic requirement, but also a question of symmetry compatibility. This perspective clarifies the assumptions underlying Bell’s theorem and highlights the role of symmetry in determining which hidden-variable models are physically realizable. This is version 1.0 of the manuscript.