Qubit exchange interactions from permutations of classical bits

Abstract

In order to prepare for the introduction of dynamical many-body and, eventually, field theoretical models, we show here that quantum mechanical exchange interactions in a three-spin chain can emerge from the deterministic dynamics of three classical Ising spins. States of the latter form an ontological basis, which will be discussed with reference to the ontology proposed in the Cellular Automaton Interpretation of Quantum Mechanics by 't Hooft. Our result illustrates a new Baker-Campbell-Hausdorff formula with terminating series expansion.

A Baker–Campbell–Hausdorff formula for the logarithm of permutations

“…Therefore, we initiated to study more complex situations in Ref. [1], namely suitable composites of Ising spins, preparing for future generalizations at present.…”

“…More generally, however, we also would like to know the representation of the Hamiltonian corresponding to the standard form ofÛ N , given in Eq. (1).…”

“…We point out that even small perturbations of the Hamiltonian, generally, will move the dynamical description from being classical, despite its appearance, to becoming the one of a quantum system [1]. Which poses interesting questions left for future work.…”

“…Our aim in this letter is to extend to the four-spin case the previous study of a chain of three classical Ising spins [1].…”

“…We do not enter the discussion of important implications for the foundations of quantum theory here -see Ref. [1] and, foremost, the book of Ref. [2] for this, including numerous references to explorations of various other "pre-quantum" models.…”