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

TL;DR

This paper demonstrates that as position measurements become perfectly precise, the probability of detecting the particle in any specific state tends to zero, implying the need for a new quantum state concept beyond Hilbert space.

Contribution

It introduces a paradox showing that perfect position measurement leads to a state outside traditional Hilbert space formalism.

Findings

Probability of finding the particle in a specific state tends to zero with perfect position measurement.

Standard density matrices cannot describe the post-measurement state in this limit.

Suggests the necessity of a new quantum state framework beyond 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.