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

Source abstract

Abstract 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 $$|\phi \rangle \langle \phi |$$ | ϕ ⟩ ⟨ ϕ | with some arbitrary but fixed normalized $$\phi$$ ϕ . Call the outcomes $$X \in [0,1]$$ X ∈ [ 0 , 1 ] and $$Y \in \{0,1\}$$ Y ∈ { 0 , 1 } . We show that in the limit $$n\rightarrow \infty$$ n → ∞ corresponding to perfect precision for X , the probability of $$Y=1$$ Y = 1 tends to 0 for every $$\phi$$ ϕ . Since there is no density matrix, pure or mixed, which upon measurement of any $$|\phi \rangle \langle \phi |$$ | 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.