Information transport in classical statistical systems

Abstract

For "static memory materials" the bulk properties depend on boundary conditions. Such materials can be realized by classical statistical systems which admit no unique equilibrium state. We describe the propagation of information from the boundary to the bulk by classical wave functions. The dependence of wave functions on the location of hypersurfaces in the bulk is governed by a linear evolution equation that can be viewed as a generalized Schrödinger equation. Classical wave functions obey the superposition principle, with local probabilities realized as bilinears of wave functions. For static memory materials the evolution within a subsector is unitary, as characteristic for the time evolution in quantum mechanics. The space-dependence in static memory materials can be used as an analogue representation of the time evolution in quantum mechanics -such materials are "quantum simulators". For example, an asymmetric Ising model on a Euclidean two-dimensional lattice represents the time evolution of free relativistic fermions in two-dimensional Minkowski space. Ising models for massless relativistic fermions 227.

Qubit exchange interactions from permutations of classical bits

“…11 in particular in the presence of interactions. 1,9,10,11,24,25 Which motivates the study of the model of Section 3.…”

“…1,4,10,11 for further discussions, as well as to other related attempts. 12,13,14,15,16,17,18,19,20,21,22,23,24,25 It may be useful to distinguish here between going beyond, as indicated, and recent reconstructions from various alternative sets of axioms, without changing the contents of quantum theory, see, e.g., Refs. 26,27,28,29 .…”

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

“…The possible relevance of Ising models for the understanding of Bell's theorem and statistical issues in quantum theory, other than serving as our model for the evolution of OS, has recently also been discussed [4,5,6,7].…”

Quantum computing with classical bits

“…The appropriate theoretical framework for the transport of probabilistic information is the quantum formalism for classical statistics [28,29]. It can be based on the "classical density matrix" ρ (t), whose diagonal elements are the "local" probabilities p(t) at t. In contrast to the local probabilities, the change from ρ (t) to a neighboring layer ρ (t + ) obeys a simple linear evolution law ρ (t + ) = S(t) ρ (t) S −1 (t) .…”

“…Here S(t) is the step evolution operator at t, which corresponds to a particular normalization of the transfer matrix [31][32][33]. In the occupation number basis the step evolution operator for generalized Ising models is a real non-negative matrix [28,29]. This restriction may be removed for more general forms of probabilistic computing.…”