Can One Detect Whether a Wave Function Has Collapsed?

TL;DR

This paper investigates the fundamental limits of detecting wave function collapse in quantum systems, showing that under various conditions, no experiment can reliably distinguish collapsed states from uncollapsed ones, especially when the initial state is unknown.

Contribution

The paper provides theoretical bounds on the ability to detect wave function collapse, demonstrating limitations in quantum measurement for different knowledge scenarios of the initial state.

Findings

No experiment can reliably detect collapse when initial state is unknown.

Detection performance is limited and no better than random guessing in the worst case.

Results depend on the prior knowledge about the initial wave function.

Abstract

Consider a quantum system prepared in state $\psi$, a unit vector in a $d$-dimensional Hilbert space. Let $b_1,...,b_d$ be an orthonormal basis and suppose that, with some probability $0<p<1$, $\psi$ ``collapses,'' i.e., gets replaced by $b_k$ (possibly times a phase factor) with Born's probability $|\langle b_k|\psi\rangle|^2$. The question we investigate is: How well can any quantum experiment on the system determine afterwards whether a collapse has occurred? The answer depends on how much is known about the initial vector $\psi$. We provide a number of different results addressing several variants of the question. In each case, no experiment can provide more than rather limited probabilistic information. In case $\psi$ is drawn randomly with uniform distribution over the unit sphere in Hilbert space, no experiment performs better than a blind guess without measurement; that is, no experiment provides any useful information.