Detecting Wave Function Collapse Without Prior Knowledge

We are concerned with the problem of detecting with high probability whether a wave function has collapsed or not, in the following framework: A quantum system with a $d$-dimensional Hilbert space is initially in state $\psi$; with probability $0<p<1$, the state collapses relative to the orthonormal basis $b_1,...,b_d$. That is, the final state $\psi'$ is random; it is $\psi$ with probability $1-p$ and $b_k$ (up to a phase) with $p$ times Born's probability $|\langle b_k|\psi \rangle|^2$. Now an experiment on the system in state $\psi'$ is desired that provides information about whether or not a collapse has occurred. Elsewhere, we identify and discuss the optimal experiment in case that $\psi$ is either known or random with a known probability distribution. Here we present results about the case that no a priori information about $\psi$ is available, while we regard $p$ and $b_1,...,b_d$ as known. For certain values of $p$, we show that the set of $\psi$s for which any experiment E is more reliable than blind guessing is at most half the unit sphere; thus, in this regime, any experiment is of questionable use, if any at all. Remarkably, however, there are other values of $p$ and experiments E such that the set of $\psi$s for which E is more reliable than blind guessing has measure greater than half the sphere, though with a conjectured maximum of 64% of the sphere.