GAP Measures and Wave Function Collapse

TL;DR

This paper explores the properties of GAP measures in quantum mechanics, demonstrating that wave functions initially distributed according to GAP measures remain GAP distributed after collapse, applicable to various measurement and collapse models.

Contribution

It proves that GAP measures are preserved under wave function collapse, a property not previously recognized, enhancing understanding of quantum state distributions.

Findings

GAP measures are preserved after wave function collapse.

The property applies to both measurement-induced and spontaneous collapses.

Provides a new perspective on quantum state distributions post-collapse.

Abstract

GAP measures (also known as Scrooge measures) are a natural class of probability distributions on the unit sphere of a Hilbert space that come up in quantum statistical mechanics; for each density matrix $\rho$ there is a unique measure GAP$_\rho$. We describe and prove a property of these measures that was not recognized so far: If a wave function $\Psi$ is GAP$_\rho$ distributed and a collapse occurs, then the collapsed wave function $\Psi'$ is again GAP distributed (relative to the appropriate $\rho'$). This fact applies to collapses due to a quantum measurement carried out by an observer, as well as to spontaneous collapse theories such as CSL or GRW. More precisely, it is the conditional distribution of $\Psi'$, given the measurement outcome (respectively, the noise in CSL or the collapse history in GRW), that is GAP$_{\rho'}$.