The framework builds on a tradition that stretches back over a century. In the 1920s, Theodor Kaluza and Oskar Klein showed that adding a compact, curled-up fifth dimension to general relativity could reproduce electromagnetism when the extra dimension is projected away. The key insight, echoed across higher-dimensional theories ever since, is that projection from a richer space into a smaller one discards information, and that discarded information can manifest as effective properties, forces, and behaviors that have no direct counterpart in the underlying description. Barros extends this logic to the heart of quantum theory. Physical states, in the new model, live on a five-dimensional manifold consisting of standard spacetime multiplied by a compact circle, parametrized by an angular coordinate that runs from zero to two pi. Observable states are obtained by integrating over this hidden angle, a projection that acts on probability amplitudes rather than probabilities, thereby preserving phase coherence across the compact dimension.

The mathematical heart of the proposal lies in a simple but powerful observation: the projection operator is non-injective, meaning that many distinct higher-dimensional configurations map onto the same observable state. The author proves a proposition showing that a projected bipartite state factorizes into independent parts only under highly constrained conditions, namely when the dependence on the compact coordinate itself separates multiplicatively between the two subsystems. Generically, this condition fails. When it fails, the projected state is non-factorizable: it becomes entangled. In other words, correlations between distant particles emerge as a structural consequence of dimensional reduction, even though the underlying higher-dimensional state was perfectly separable. Entanglement, in this picture, is not a primitive ingredient of reality but an effective phenomenon, akin to how an effective electromagnetic force emerges in Kaluza-Klein theory from pure geometry.

The most striking demonstration of the mechanism is the construction of a Bell state from scratch. Barros considers a two-level system, such as photon polarization with horizontal and vertical states, and writes down a state in the extended space that is manifestly separable: each subsystem carries its own coherent superposition with phases that depend on the hidden angular coordinate. When the projection is applied, the Fourier structure of the compact dimension does the rest. Terms in the tensor product carrying nonzero phase harmonics, such as factors of e to the two-i-theta, integrate to zero by the orthogonality of Fourier modes. Only the phase-neutral terms survive, and after normalization the resulting observable state is exactly the maximally entangled Bell state, with horizontal-horizontal and vertical-vertical components in equal superposition. The interference along the hidden circle, followed by dimensional reduction, manufactures entanglement from a state that possessed none.

Crucially, the framework is not merely a reinterpretation of existing quantum mechanics; it makes testable predictions. When phase cancellation along the compact dimension is incomplete, characterized by a small parameter eta, the projected state acquires additional components that modify the observable correlations. The standard quantum prediction for maximally entangled states is that the correlation function depends only on the difference between the two measurement angles, giving the familiar cosine of a minus b. In the modified framework, an extra term proportional to the cosine of a plus b appears, weighted by a parameter epsilon that measures the residual influence of higher-dimensional modes. This is a qualitatively distinct signature. Ordinary experimental imperfections, such as reduced visibility, simply rescale the standard cosine dependence, whereas the projection-induced term has a different angular structure entirely, meaning that sufficiently precise measurements can in principle distinguish the two.

The author shows that the deviation parameter epsilon could be extracted directly from experiment. By choosing measurement settings such that the standard quantum contribution vanishes, for example with analyzer angles separated by three quarters of pi, the correlation function reduces to exactly epsilon, allowing a single measurement to reveal the presence or absence of higher-dimensional effects. More generally, the full two-parameter angular landscape of correlations can be mapped and fitted to separate the cosine of a minus b from the cosine of a plus b contributions. The statistical requirements are demanding but well within reach of modern photonic platforms. Resolving a deviation of one percent at the three-sigma confidence level requires roughly one hundred thousand detected coincidence events, while a four percent deviation needs only about ten thousand. State-of-the-art Bell experiments routinely achieve pair-generation rates and interference visibilities above ninety-nine percent, placing such measurements squarely within current technological capability.

An important conceptual point is that the model does not fall afoul of Bell’s theorem. The compact coordinate is not an accessible classical hidden variable that predetermines individual measurement outcomes, which is the scenario that Bell’s theorem and its experimental tests rule out. Instead, the projection acts coherently on probability amplitudes, and the resulting correlations remain incompatible with the CHSH bound: the author demonstrates that for the standard optimal measurement settings, the epsilon-dependent contribution cancels exactly in the CHSH combination, so the inequality remains violated throughout the parameter domain. The apparent nonlocality of quantum correlations is thus reinterpreted as an effective phenomenon. Observers confined to the projected spacetime see correlations that cannot be explained locally, while the underlying higher-dimensional description remains local through its dependence on a shared compact coordinate. No superluminal signaling is required, and the framework remains compatible with the no-signalling principle, since the correlations are fixed by the global projected state before any measurement takes place.

The work is explicitly limited to a kinematic level, describing the structure of states and their projection without yet specifying how the higher-dimensional state evolves in time. The author sketches a natural dynamical extension, in which a Hamiltonian defined on the extended space induces effective evolution in the observable domain through the projection operator, potentially explaining the origin of unitary quantum dynamics in geometric terms. Whether the harmonic structure along the compact dimension remains stable under such dynamics, preserving the phase-coherent mode pairs on which the entanglement mechanism depends, is identified as a central open problem for future work. A complete derivation of the effective evolution equation, and a quantitative comparison of the predicted epsilon bounds with the uncertainties of existing high-precision Bell experiments, remain to be carried out.

Even in its present form, the proposal carries considerable conceptual weight. It suggests that several hallmark features of quantum theory, from interference to entropy to entanglement, might share a common geometric origin in the non-injective mapping between a higher-dimensional description and its observable projection. If future experiments detect a nonzero epsilon, it would constitute direct evidence that observable correlations retain imprints of degrees of freedom beyond standard spacetime, a result that would rank among the most profound discoveries in physics. Conversely, increasingly stringent experimental bounds on epsilon would constrain the class of admissible higher-dimensional states, turning the framework into a progressively sharper instrument for probing the geometry underlying quantum mechanics. Either outcome advances the centuries-old project of understanding why the quantum world looks the way it does, and whether the strangeness of entanglement is a fundamental law or, as this work suggests, a shadow cast by dimensions we cannot see.

Subject of Research: Emergence of quantum entanglement from geometric projection of higher-dimensional states

Article Title: Emergent quantum entanglement from higher-dimensional geometric projection

Article References: Barros, A. K. (2026). Emergent quantum entanglement from higher-dimensional geometric projection. Results in Physics, Article 108761. https://doi.org/10.1016/j.rinp.2026.108761

Image Credits: AI Generated

DOI: 10.1016/j.rinp.2026.108761

Keywords: quantum entanglement, higher dimensions, Kaluza-Klein theory, Bell inequality, quantum foundations, dimensional projection, compact dimension, CHSH bound, photonic experiments, quantum correlations, nonlocality, theoretical physics