Marek Czachor

A hierarchical structure of isomorphic arithmetics is defined by a bijection gR : R → R. It entails a hierarchy of probabilistic models, with probabilities pk = gk(p), where g is the restriction of gR to the interval [0, 1], gk is the kth iterate of g, and k is an arbitrary integer (positive, negative, or zero; g0(x) = x). The relation between p and gk(p), k > 0, is analogous to the one between probability and neural activation function. For k ≪ −1, gk(p) is essentially a white noise (all processes are equally probable). The choice of k = 0 is physically as arbitrary as the choice of origin of a line in space, hence what we regard as experimental binary probabilities, pexp, can be given by any k, pexp = gk(p). Quantum binary probabilities are defined by g(p) = sin2 π2 p. With this concrete form of g, one finds that any two neighboring levels of the hierarchy are related to each other in a quantum-subquantum relation. In this sense, any model in the hierarchy is probabilistically quantum in appropriate arithmetic and calculus. And the other way around: any model is subquantum in appropriate arithmetic and calculus. Probabilities involving more than two events are constructed by means of trees of binary conditional probabilities. We discuss from this perspective singlet-state probabilities and Bell inequalities. We find that singlet state probabilities involve simultaneously three levels of the hierarchy: quantum, hidden, and macroscopic. As a by-product of the analysis we discover a new (arithmetic) interpretation of the Fubini-Study geodesic distance.

Any single system whose space of states is given by a separable Hilbert space is automatically equipped with infinitely many hidden tensor-like structures. This includes all quantum mechanical systems as well as classical field theories and classical signal analysis. Accordingly, systems as simple as a single one-dimensional harmonic oscillator, an infinite potential well, or a classical finite-amplitude signal of finite duration can be decomposed into an arbitrary number of subsystems. The resulting structure is rich enough to enable quantum computation, violation of Bell’s inequalities, and formulation of universal quantum gates. Less standard quantum applications involve a distinction between position and hidden position. The hidden position can be accompanied by a hidden spin, even if the particle is spinless. Hidden degrees of freedom are, in many respects, analogous to modular variables. Moreover, it is shown that these hidden structures are at the roots of some well-known theoretical constructions, such as the Brandt–Greenberg multi-boson representation of creation–annihilation operators, intensively investigated in the context of higher-order or fractional-order squeezing. In the context of classical signal analysis, the discussed structures explain why it is possible to emulate a quantum computer by classical analog circuit devices.

In this letter, we demonstrate that the fractional curl operator, widely used in electromagnetics since 1998, is essentially a rotation operation of components of the complex Riemann-Silberstein vector representing the electromagnetic field. It occurs that after the wave decomposition into circular polarisations, the standard duality rotation with the angle depending on the fractional order is applied to the left-handed basis vector whilst the right-handed basis vector stems from the complex conjugation of the left-handed counterpart. Therefore, the fractional curl operator describes another representation of rotations of the electromagnetic field decomposed into circular polarisations. Finally, we demonstrate that this operator can describe a single-qubit phase-shift gate in quantum computing.