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 $ψ$; with probability $0<p<1$, the state collapses relative to the orthonormal basis $b_1,...,b_d$. That is, the final state $ψ'$ is random; it is $ψ$ with probability $1-p$ and $b_k$ (up to a phase) with $p$ times Born's probability $|\langle b_k|ψ\rangle|^2$. Now an experiment on the system in state $ψ'$ is desired that provides information about whether or not a collapse has occurred. Elsewhere, we identify and discuss the optimal experiment in case that $ψ$ is either known or random with a known probability distribution. Here we present results about the case that no a priori information about $ψ$ is available, while we regard $p$ and $b_1,...,b_d$ as known. For certain values of $p$, we show that the set of $ψ$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 $ψ$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.

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