Quantum Zeno-like Paradox for Position Measurements: A Particle Precisely Found in Space is Nowhere to be Found in Hilbert Space
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910032404152320 |
|---|---|
| author | Oianguren-Asua, Xabier Tumulka, Roderich |
| author_facet | Oianguren-Asua, Xabier Tumulka, Roderich |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19469 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum Zeno-like Paradox for Position Measurements: A Particle Precisely Found in Space is Nowhere to be Found in Hilbert Space Oianguren-Asua, Xabier Tumulka, Roderich Quantum Physics Mathematical Physics 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. |
| title | Quantum Zeno-like Paradox for Position Measurements: A Particle Precisely Found in Space is Nowhere to be Found in Hilbert Space |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2601.19469 |