Perfect quantum protractors

Michał Piotrak; Marek Kopciuch; Arash Dezhang Fard; Magdalena Smolis; Szymon Pustelny; Kamil Korzekwa

doi:10.22331/q-2024-09-03-1459

Perfect quantum protractors

Michał Piotrak^1,2, Marek Kopciuch^3,4, Arash Dezhang Fard^3,4, Magdalena Smolis⁴, Szymon Pustelny⁴, and Kamil Korzekwa⁴

¹Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom
²Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom
³Doctoral School of Exact and Natural Sciences, Jagiellonian University, Faculty of Physics, Astronomy and Applied Computer Sciences, 30-348 Kraków, Poland
⁴Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Kraków, Poland

Published:	2024-09-03, volume 8, page 1459
Eprint:	arXiv:2310.13045v2
Doi:	https://doi.org/10.22331/q-2024-09-03-1459
Citation:	Quantum 8, 1459 (2024).

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Abstract

In this paper we introduce and investigate the concept of a $\textit{perfect quantum protractor}$, a pure quantum state $|\psi\rangle\in\mathcal{H}$ that generates three different orthogonal bases of $\mathcal{H}$ under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variance-based measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a well-defined total angular momentum $j$, and we prove that they do not exist for $j\in\{1/2,2,5/2\}$, but they do exist for $j\in\{1,3/2,3\}$ (with numerical evidence for their existence when $j=7/2$). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not $\textit{a priori}$ known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium-87, where we prepare a perfect quantum protractor for a spin-1 system, let it precess around $x$, $y$ or $z$ axis, and then employ it to optimally estimate the rotation angle.

Featured image: Experimental setup for preparing a perfect quantum protractor with warm atomic vapours of rubidium-87.

► BibTeX data

@article{Piotrak2024perfectquantum,
  doi = {10.22331/q-2024-09-03-1459},
  url = {https://doi.org/10.22331/q-2024-09-03-1459},
  title = {Perfect quantum protractors},
  author = {Piotrak, Micha{\l{}} and Kopciuch, Marek and Fard, Arash Dezhang and Smolis, Magdalena and Pustelny, Szymon and Korzekwa, Kamil},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1459},
  month = sep,
  year = {2024}
}

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[1] Shaowei Du, Shuheng Liu, Frank E. S. Steinhoff, and Giuseppe Vitagliano, "Characterizing resources for multiparameter estimation of SU(2) and SU(1,1) unitaries", Quantum 10, 2130 (2026).

[2] Arash Dezhang Fard, Marek Kopciuch, Yujie Sun, Przemysław Włodarczyk, and Szymon Pustelny, "Isolating pure quadratic Zeeman splitting", Physical Review Applied 23 6, 064034 (2025).

[3] Matthew J. Lake and Marek Miller, "Quantum reference frames, revisited", arXiv:2312.03811, (2023).

[4] Marcin Rudziński, Adam Burchardt, and Karol Życzkowski, "Orthonormal bases of extreme quantumness", Quantum 8, 1234 (2024).

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