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Can One Detect Whether a Wave Function Has Collapsed?

Consider a quantum system prepared in state $ψ$, a unit vector in a $d$-dimensional Hilbert space. Let $b_1,...,b_d$ be an orthonormal basis and suppose that, with some probability $0<p<1$, $ψ$ ``collapses,'' i.e., gets replaced by $b_k$ (possibly times a phase factor) with Born's probability $|\langle b_k|ψ\rangle|^2$. 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 $ψ$. 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 $ψ$ 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.