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Attainability of the Heisenberg limit for the estimation of the rotation angle is demonstrated for maximal squeezing. For a specific direction and strength an advantage in sensitivity for a<\/mml:mi>l<\/mml:mi>l<\/mml:mi><\/mml:math> equatorial rotation axes (and hence non-commuting rotations) over the classical bound is shown in terms of quadratic scaling of the single-parameter quantum Fisher information for the corresponding rotation angles. Our results provide a method for measuring magnetic fields in any direction in the x<\/mml:mi><\/mml:math>-y<\/mml:mi><\/mml:math>-plane with the same optimized initial state.<\/jats:p>","DOI":"10.22331\/q-2024-05-08-1341","type":"journal-article","created":{"date-parts":[[2024,5,8]],"date-time":"2024-05-08T15:04:55Z","timestamp":1715180695000},"page":"1341","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Minimal-noise estimation of noncommuting rotations of a spin"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4062-833X","authenticated-orcid":false,"given":"Jakub","family":"Czartowski","sequence":"first","affiliation":[{"name":"Doctoral School of Exact and Natural Sciences, Jagiellonian University, ul. \u0141ojasiewicza 11, 30-348 Krak\u00f3w, Poland"},{"name":"Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. \u0141ojasiewicza 11, 30-348 Krak\u00f3w, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0653-3639","authenticated-orcid":false,"given":"Karol","family":"\u017byczkowski","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. \u0141ojasiewicza 11, 30-348 Krak\u00f3w, Poland"},{"name":"Centrum Fizyki Teoretycznej PAN, Al. Lotnik\u00f3w 32\/46, 02-668 Warszawa, Poland"},{"name":"National Quantum Information Center (KCIK), University of Gda\u0144sk, Poland"}]},{"given":"Daniel","family":"Braun","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics, University of T\u00fcbingen, Auf der Morgenstelle 14, 72076 T\u00fcbingen, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2024,5,8]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"R. Demkowicz-Dobrza\u0144ski, K. Banaszek, and R. Schnabel. ``Fundamental quantum interferometry bound for the squeezed-light-enhanced gravitational wave detector GEO 600''. Phys. Rev. A 88, 041802 (2013).","DOI":"10.1103\/PhysRevA.88.041802"},{"key":"1","doi-asserted-by":"publisher","unstructured":"J. Aasi et al. ``Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light''. 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