Dendrogramic Representation of Data: CHSH Violation vs. Nonergodicity

Entropy (Basel). 2021 Jul 28;23(8):971. doi: 10.3390/e23080971.

Abstract

This paper is devoted to the foundational problems of dendrogramic holographic theory (DH theory). We used the ontic-epistemic (implicate-explicate order) methodology. The epistemic counterpart is based on the representation of data by dendrograms constructed with hierarchic clustering algorithms. The ontic universe is described as a p-adic tree; it is zero-dimensional, totally disconnected, disordered, and bounded (in p-adic ultrametric spaces). Classical-quantum interrelations lose their sharpness; generally, simple dendrograms are "more quantum" than complex ones. We used the CHSH inequality as a measure of quantum-likeness. We demonstrate that it can be violated by classical experimental data represented by dendrograms. The seed of this violation is neither nonlocality nor a rejection of realism, but the nonergodicity of dendrogramic time series. Generally, the violation of ergodicity is one of the basic features of DH theory. The dendrogramic representation leads to the local realistic model that violates the CHSH inequality. We also considered DH theory for Minkowski geometry and monitored the dependence of CHSH violation and nonergodicity on geometry, as well as a Lorentz transformation of data.

Keywords: CHSH inequality; clustering algorithms; dendrogramic holographic theory; dendrograms; epistemic; nonergodicity; ontic; quantumness.