A statistical model for quantum spin and photon number states
The most irreducible way to represent information is a sequence of two symbols. In this paper, we construct quantum states using this basic building block. Specifically, we show that the probabilities that arise in quantum theory can be reduced to counting more fundamental ontic states, which we interpret as event networks and model using sequences of 0's and 1's. A completely self contained formalism is developed for the purpose of organizing and counting these ontic states, which employs the finite cyclic group $\mathbb{Z}_2 = \{0, 1\}$, basic set theory, and combinatorics. This formalism is then used to calculate probability distributions associated with particles of arbitrary spin interacting with sequences of two rotated Stern-Gerlach detectors. A central ingredient of this construction is the rule which converts the abstract counts labelling an operation into the physical angle of rotation it represents. We show that this rule is not linear in the counts, but is instead fixed by the half-angle law $\tan(θ_{ab}/2)=\tilde{B}_{map}/\tilde{A}_{map}$, which is the unique assignment consistent with the composition of successive rotations. These calculations are compared with the predictions of non-relativistic quantum mechanics and shown to agree exactly in the limit of large sequence length $n$, with finite $n$ corrections which vanish as $O(1/n)$ and with no free or fitted parameters. The residual deviation at finite $n$ does not lead to violations of relevant no-go theorems, such as Bell's inequalities, the Kochen-Specker theorem, or the PBR theorem. The proposed model is then extended to an optical system involving photon number states passing through a beam splitter. Leveraging recent advancements in high precision experiments on these systems, we then propose a means of testing the new model using a tabletop experiment.