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GAP Measures and Wave Function Collapse

Roderich Tumulka

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 $ρ$ there is a unique measure GAP$_ρ$. We describe and prove a property of these measures that was not recognized so far: If a wave function $Ψ$ is GAP$_ρ$ distributed and a collapse occurs, then the collapsed wave function $Ψ'$ is again GAP distributed (relative to the appropriate $ρ'$). 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 $Ψ'$, given the measurement outcome (respectively, the noise in CSL or the collapse history in GRW), that is GAP$_{ρ'}$.

GAP Measures and Wave Function 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 there is a unique measure GAP. We describe and prove a property of these measures that was not recognized so far: If a wave function is GAP distributed and a collapse occurs, then the collapsed wave function is again GAP distributed (relative to the appropriate ). 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 , given the measurement outcome (respectively, the noise in CSL or the collapse history in GRW), that is GAP.
Paper Structure (3 sections, 1 theorem, 17 equations)

This paper contains 3 sections, 1 theorem, 17 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Examples

Key Result

Theorem 1

Let $\mathscr{H}$ be a Hilbert space and $\Psi$ a random point on $\mathbb{S}(\mathscr{H})$ (the "initial wave function"). Suppose for each point $x$ in the measure space $(\mathscr{X},\mathscr{F},\mu)$, we are given an operator $L(x)$ ("collapse operator") such that Suppose the random point $X$ gets chosen with probability distribution ("Born distribution"), and define (the "final wave functio

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2