In the contemporary age of everincreasing influence of electronic media as much as on an individual psyche as on the society as a whole, it would be of great value to undertake a detailed analysis of the very foundation of that means of communication: the electron. This means also re-evaluating the notions of modern physics on the nature of matter, most importantly the notions of the quantum field theory and general theory of relativity. Some of the questions discussed are: what is an electron, what does it mean to say it is an elementary particle and, in general, what is the meaning of the notion of the elementary particle? Also, how are signals transmitted via particles, so how is telecommunication possible, and wirelessly? What is the meaning and nature of physical interaction?
We investigate the task of identifying the parity (odd vs even) of an unknown permutation applied to n particles. Classically, using fewer than n distinct labels per particle limits the success probability to random guessing, whereas quantum mechanics, exploiting entanglement in both preparation and measurement, accomplishes the task perfectly with as few as ⌈n⌉ levels per particle [\href{https://doi.org/10.1103/yhyv-xnwq}{PRL {\bf 135}, 260603 (2025)}]. We show that even without entangled preparation, quantum theory still offers a probabilistic advantage over classical strategies. Moreover, such product preparations yield perfect success in locally quantum theories, where elementary systems are quantum but their composition follows the minimal tensor product structure of generalized probabilistic theories (GPTs). We further identify GPT models that accomplish the task with certainty without requiring entanglement either at the preparation stage or at the measurement stage. Our central result establishes that the linear dimension of the elementary systems, rather than entanglement, is the fundamental resource governing the existence of probabilistic advantage in the permutation parity problem. In particular, below the required dimension threshold, no amount of entanglement can improve upon the random-guessing limit.
In the operational approach to general probabilistic theories one distinguishes two spaces, the state space of the "elementary systems" and the physical space in which "laboratory devices" are embedded. Each of those spaces has its own dimension- the minimal number of real parameters (coordinates) needed to specify the state of system or a point within the physical space. Within an operational framework to a physical theory, the two dimensions coincide in a natural way under the following "closeness" requirement: the dynamics of a single elementary system can be generated by the invariant interaction between the system and the "macroscopic transformation device" that itself is described from within the theory in the macroscopic (classical) limit. Quantum mechanics fulfils this requirement since an arbitrary unitary transformation of an elementary system (spin-1/2 or qubit) can be generated by the pairwise invariant interaction between the spin and the constituents of a large coherent state ("classical magnetic field"). Both the spin state space and the "classical field" are then embedded in the Euclidean three-dimensional space. Can we have a general probabilistic theory, other than quantum theory, in which the elementary system ("generalized spin") and the "classical fields" generating its dynamics are embedded in a higher-dimensional physical space? We show that as long as the interaction is pairwise, this is impossible, and quantum mechanics and the three-dimensional space remain the only solution. However, having multi-particle interactions and a generalized notion of "classical field" may open up such a possibility.