# Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States

John Martin1, Stefan Weigert2, and Olivier Giraud3

1Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, University of Liège, B-4000 Liège, Belgium
2Department of Mathematics, University of York, UK-York YO10 5DD, United Kingdom
3Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France

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### Abstract

Coherent and anticoherent states of spin systems up to spin $j=2$ are known to be optimal in order to detect rotations by a known angle but unknown rotation axis. These optimal quantum rotosensors are characterized by minimal fidelity, given by the overlap of a state before and after a rotation, averaged over all directions in space. We calculate a closed-form expression for the average fidelity in terms of anticoherent measures, valid for arbitrary values of the quantum number $j$. We identify optimal rotosensors (i) for arbitrary rotation angles in the case of spin quantum numbers up to $j=7/2$ and (ii) for small rotation angles in the case of spin quantum numbers up to $j=5$. The closed-form expression we derive allows us to explain the central role of anticoherence measures in the problem of optimal detection of rotation angles for arbitrary values of $j$.

Advances in measurement techniques have often led to progress in physics. Over time, metrology developed as a subject of its own, especially in the context of defining standard units for physical quantities. Quantum theory provides new perspectives on measurements but also new challenges. The currently emerging quantum technologies require ever better control of microscopic systems and, hence, measurements which are as accurate as possible. In this paper, we are interested to determine whether a quantum system has undergone a rotation by a known angle about an unknown axis. The states optimally suited for this task are called "optimal quantum rotosensors". They are characterized by minimal fidelity, given by the overlap of a state before and after a rotation, averaged over all directions in space. The closed-form expression we derive for the fidelity and numerical computations allow us to explain the central role of a specific class of quantum states in this problem which are known as "anticoherent" states.

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### Cited by

[1] Aaron Z Goldberg, Andrei B Klimov, Gerd Leuchs, and Luis L Sánchez-Soto, "Rotation sensing at the ultimate limit", Journal of Physics: Photonics 3 2, 022008 (2021).

[2] Aaron Z. Goldberg, Andrei B. Klimov, Markus Grassl, Gerd Leuchs, and Luis L. Sánchez-Soto, "Extremal quantum states", AVS Quantum Science 2 4, 044701 (2020).

[3] Aaron Z. Goldberg, Pablo de la Hoz, Gunnar Björk, Andrei B. Klimov, Markus Grassl, Gerd Leuchs, and Luis L. Sánchez-Soto, "Quantum concepts in optical polarization", Advances in Optics and Photonics 13 1, 1 (2021).

[4] Naeem Akhtar, Barry C. Sanders, and Carlos Navarrete-Benlloch, "Sub-Planck structures: Analogies between the Heisenberg-Weyl and SU(2) groups", Physical Review A 103 5, 053711 (2021).

[5] Daniel Basilewitsch, Haidong Yuan, and Christiane P. Koch, "Optimally controlled quantum discrimination and estimation", Physical Review Research 2 3, 033396 (2020).

[6] Chryssomalis Chryssomalakos, Louis Hanotel, Edgar Guzmán-González, Daniel Braun, Eduardo Serrano-Ensástiga, and Karol Życzkowski, "Symmetric multiqudit states: Stars, entanglement, and rotosensors", Physical Review A 104 1, 012407 (2021).

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