Quantum Zeno-like Paradox for Position Measurements: A Particle Precisely Found in Space is Nowhere to be Found in Hilbert Space
Xabier Oianguren-Asua, Roderich Tumulka
TL;DR
This work analyzes a spatial analogue of the quantum Zeno effect by modeling a particle in $[0,1)$ with a sequence of increasingly precise position measurements $X^{(n)}$ (bins of width $1/n$) followed by a projection measurement $Y=|\phi\rangle\langle\phi|$. It proves that $\mathbb{P}^{(n)}(Y=1)\to 0$ as $n\to\infty$ for all unit vectors, implying the post-measurement state cannot be represented by any vector or density operator in the Hilbert space $\mathcal{H}=L^2([0,1))$. The result motivates the existence of a novel quantum state concept beyond Hilbert space and extends to mixed states and higher dimensions. The findings connect with continuous observables, contrast with the time-based quantum Zeno effect, and lay groundwork for exploring a state that acts on operator algebras to describe the limiting procedure.
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 $|φ\rangle\langleφ|$ with some arbitrary but fixed normalized $φ$. 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 $φ$. Since there is no density matrix, pure or mixed, which upon measurement of any $|φ\rangle\langleφ|$ 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.