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 inaccuracy $1\mathrm{/}n$ and then a quantum measurement of the projection $\mathrm{\mid }\phi ⟩⟨\phi \mathrm{\mid }$ with some arbitrary but fixed normalized $\phi$. Call the outcomes $X\in [0,1]$ and $Y\in \{0,1\}$. We show that in the limit $n\to \mathrm{\infty }$ corresponding to perfect precision for $X$, the probability of $Y=1$ tends to 0 for every $\phi$. Since there is no density matrix, pure or mixed, which upon measurement of any $\mathrm{\mid }\phi ⟩⟨\phi \mathrm{\mid }$ 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.