The collapse of a quantum state as a joint probability construction

TL;DR

This paper introduces a mathematical framework for quantum state collapse as a joint probability construction, using a non-commutative, non-associative product, offering an alternative to the traditional collapse interpretation.

Contribution

It formalizes the collapse process as a nonlinear, non-commutative product and presents a no-collapse model using L"uders transformers for joint measurements.

Findings

Collapse as a joint probability construction

Equivalent no-collapse model with L"uders transformers

Simplified analysis for large measurement sequences

Abstract

The collapse of a quantum state can be understood as a mathematical way to construct a joint probability density even for operators that do not commute. We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at timelike separation, in part inspired by the sequential product for positive semi-definite operators. The familiar collapse picture, in which a quantum state collapses after each measurement as a way to construct a joint probability density for consecutive measurements, is equivalent to a no-collapse picture in which L\"uders transformers applied to subsequent measurements construct a Quantum-Mechanics--Free-Subsystem of Quantum Non-Demolition operators, not as a dynamical process but as an alternative mathematical model for the same consecutive measurements. The no-collapse picture is particularly simpler when we apply signal analysis to millions or billions of consecutive measurements.