{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T13:16:06Z","timestamp":1754486166600},"reference-count":41,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T00:00:00Z","timestamp":1725321600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001870","name":"Foundation for Polish Science","doi-asserted-by":"crossref","award":["POIR.04.04.00-00-17C1\/18-00"],"award-info":[{"award-number":["POIR.04.04.00-00-17C1\/18-00"]}],"id":[{"id":"10.13039\/501100001870","id-type":"DOI","asserted-by":"crossref"}]},{"name":"National Science Centre, Poland","award":["2019\/34\/E\/ST2\/00440"],"award-info":[{"award-number":["2019\/34\/E\/ST2\/00440"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"In this paper we introduce and investigate the concept of a perfect quantum protractor<\/mml:mtext><\/mml:mrow><\/mml:math>, a pure quantum state |<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>&#x27E9;<\/mml:mo>&#x2208;<\/mml:mo>H<\/mml:mi><\/mml:mrow><\/mml:math> that generates three different orthogonal bases of H<\/mml:mi><\/mml:mrow><\/mml:math> 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<\/mml:mi><\/mml:math>, and we prove that they do not exist for j<\/mml:mi>&#x2208;<\/mml:mo>{<\/mml:mo>1<\/mml:mn>\/<\/mml:mo><\/mml:mrow>2<\/mml:mn>,<\/mml:mo>2<\/mml:mn>,<\/mml:mo>5<\/mml:mn>\/<\/mml:mo><\/mml:mrow>2<\/mml:mn>}<\/mml:mo><\/mml:math>, but they do exist for j<\/mml:mi>&#x2208;<\/mml:mo>{<\/mml:mo>1<\/mml:mn>,<\/mml:mo>3<\/mml:mn>\/<\/mml:mo><\/mml:mrow>2<\/mml:mn>,<\/mml:mo>3<\/mml:mn>}<\/mml:mo><\/mml:math> (with numerical evidence for their existence when j<\/mml:mi>=<\/mml:mo>7<\/mml:mn>\/<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:math>). 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 a priori<\/mml:mtext><\/mml:mrow><\/mml:math> 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<\/mml:mi><\/mml:math>, y<\/mml:mi><\/mml:math> or z<\/mml:mi><\/mml:math> axis, and then employ it to optimally estimate the rotation angle.<\/jats:p>","DOI":"10.22331\/q-2024-09-03-1459","type":"journal-article","created":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T12:50:17Z","timestamp":1725367817000},"page":"1459","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Perfect quantum protractors"],"prefix":"10.22331","volume":"8","author":[{"given":"Micha\u0142","family":"Piotrak","sequence":"first","affiliation":[{"name":"Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom"},{"name":"Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom"}]},{"given":"Marek","family":"Kopciuch","sequence":"additional","affiliation":[{"name":"Doctoral School of Exact and Natural Sciences, Jagiellonian University, Faculty of Physics, Astronomy and Applied Computer Sciences, 30-348 Krak\u00f3w, Poland"},{"name":"Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Krak\u00f3w, Poland"}]},{"given":"Arash Dezhang","family":"Fard","sequence":"additional","affiliation":[{"name":"Doctoral School of Exact and Natural Sciences, Jagiellonian University, Faculty of Physics, Astronomy and Applied Computer Sciences, 30-348 Krak\u00f3w, Poland"},{"name":"Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Krak\u00f3w, Poland"}]},{"given":"Magdalena","family":"Smolis","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Krak\u00f3w, Poland"}]},{"given":"Szymon","family":"Pustelny","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Krak\u00f3w, Poland"}]},{"given":"Kamil","family":"Korzekwa","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Krak\u00f3w, Poland"}]}],"member":"9598","published-online":{"date-parts":[[2024,9,3]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Werner Heisenberg. ``\u00dcber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik''. 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