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 $|\varphi ⟩⟨\varphi |$ with some arbitrary but fixed normalized $\varphi$. 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 $\varphi$. Since there is no density matrix, pure or mixed, which upon measurement of any $|\varphi ⟩⟨\varphi |$ 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.

12 pages LaTeX, no figures; v3 minor additions

References in corpus (1)

• Control of quantum noise: on the role of dilations