Explaining Bell Locally

TL;DR

This work argues that Bell inequality violations can be fully reconciled with locality within the Heisenberg-picture, unitary quantum theory by employing Deutsch–Hayden descriptors. It shows how local, autonomous world copies (foliations) arise from remote measurements and, when brought together, yield four-outcome records that reproduce CHSH-violating statistics with a win rate of $\cos^2(\pi/8)$. The analysis integrates Everettian ideas and decoherence, demonstrating that relativistic considerations and classicality concepts do not undermine locality in this framework. The result provides a realist, fully local interpretation of Bell correlations, with implications for quantum foundations and information processing.

Abstract

In the Heisenberg picture of unitary quantum theory, Bell inequalities are violated with local elements of reality interacting locally. Here is how: Upon measuring her particle of the entangled pair, Alice -- like other coupled systems -- smoothly and locally evolves into two non-interacting versions of herself, each of which records a different outcome: she foliates. Everything that suitably interacts with the Alices foliates in turn, generating worlds which, for all practical purposes, remain distinct and autonomous. At spacelike separation, an analogous yet independent process occurs to Bob when he measures his particle, locally differentiating him and his surroundings into two non-interacting instances. To confirm the violation of Bell inequalities, Alice and Bob must further interact to produce a record of the joint outcomes. The record arises from the two local worlds of Alice, and those of Bob, and foliates into four instances: '00', '01', '10' and '11'. The outcomes that win the Clauser--Horne--Shimony--Holt (CHSH) game sum to a measure of $\cos^2(π/8)$.

Figures (3)

• Figure 1: Schematic depiction of a Bell-inequality violation in the Heisenberg picture of unitary quantum mechanics. The thickness of a branch reflects its probability measure.
• Figure 2: A Bell experiment. Thanks to the locality of descriptors, the calculations are performed directly on the wires, with the help of equations \ref{['eqH']}, \ref{['eqR']} and \ref{['eqCnot']}. A descriptor with no time label indicates its initial form, which for qubits is set by equation \ref{['eq:initialdescriptor']}. The notations $s_{\gamma}$ and $c_{\gamma}$ stand for $\sin \gamma$ and $\cos \gamma$.
• Figure 3: The Bell experiment is enhanced with considerations of classicality. First, after $\mathfrak Q_1$'s rotation, but before Alice's measurement, a decoherent interaction involves an environmental system, which contains at least the logical space of a qubit. Unlike the initialized qubits, the environment's descriptor $(\bar{q}_{Ex}, \bar{q}_{Ez})$ is given by some generic representation of the Pauli algebra, which generically differ from that of equation \ref{['eq:initialdescriptor']}. Second, a chain reaction through properly initialized systems $\mathfrak Q_{A'}$, $\mathfrak Q_{A"}$—and generically many more—models classical communication. The nontrivial arrangement of $\mathfrak S_C$'s relative descriptors is unhindered by these considerations of classicality.