Quantum Zeno-like paradox for position measurements: a particle precisely found in space is nowhere to be found in Hilbert space

On a quantum particle in the unit interval [0, 1], perform a position measurement with imprecision 1/ n and then a quantum measurement of the projection |φ φ | | ϕ ⟩ ⟨ ϕ | with some arbitrary but fixed normalized φ ϕ . Call the outcomes X ∈ [0,1] X ∈ [ 0 , 1 ] and Y ∈ \0,1\ Y ∈ { 0 , 1 } . We show that in the limit n→ ∞ n → ∞ corresponding to perfect precision for X , the probability of $$Y=1$$ Y = 1 tends to 0 for every φ ϕ . Since there is no density matrix, pure or mixed, which upon measurement of any |φ φ | | ϕ ⟩ ⟨ ϕ | 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.