Abstract
On a quantum particle in the unit interval [0,1], perform a position measurement with inaccuracy 1/n and then a quantum measurement of the projection |φφ| with some arbitrary but fixed normalized φ. Call the outcomes X ∈[0,1] and Y ∈{0,1}. We show that in the limit n→∞ corresponding to perfect precision for X, the probability of Y=1 tends to 0 for every φ. Since there is no density matrix, pure or mixed, which upon measurement of any |φφ| yields outcome 1 with probability 0, our result suggests that a novel type of quantum state beyond Hilbert space is necessary to describe a quantum particle after a perfect position measurement.