Can one detect whether a wave function has collapsed?

Abstract

Consider a quantum system prepared in state ψ, a unit vector in a d-dimensional Hilbert space. Let b1, …, bd be an orthonormal basis and suppose that, with some probability 0 < p < 1, ψ ‘collapses’, i.e., gets replaced by bk (possibly times a phase factor) with Born’s probability |〈bk|ψ〉|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.