Quantum Zeno-like Paradox for Position Measurements: A Particle Precisely Found in Space is Nowhere to be Found in Hilbert Space

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 $|\phi\rangle\langle\phi|$ 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 $n\to\infty$ 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 $|\phi\rangle\langle\phi|$ 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.