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 b₁, …, 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 $b k | ψ 2$⁠. Now an experiment on the system in state ψ′ is desired that provides information about whether or not a collapse has occurred. Elsewhere [C. W. Cowan and R. Tumulka, J. Phys. A: Math. Theor. 47, 195303 (2014)], 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 b₁, …, b_d as known. For certain values of p, we show that the set of ψs 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 ψs 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.