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

John Martin; Stefan Weigert; Olivier Giraud

doi:10.22331/q-2020-06-22-285

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

Published:	2020-06-22, volume 4, page 285
Eprint:	arXiv:1909.08355v2
Doi:	https://doi.org/10.22331/q-2020-06-22-285
Citation:	Quantum 4, 285 (2020).

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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 $j$.

Popular summary

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.

► BibTeX data

@article{Martin2020optimaldetectionof,
  doi = {10.22331/q-2020-06-22-285},
  url = {https://doi.org/10.22331/q-2020-06-22-285},
  title = {Optimal {D}etection of {R}otations about {U}nknown {A}xes by {C}oherent and {A}nticoherent {S}tates},
  author = {Martin, John and Weigert, Stefan and Giraud, Olivier},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {285},
  month = jun,
  year = {2020}
}

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