{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T04:01:12Z","timestamp":1768449672248,"version":"3.49.0"},"reference-count":61,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,8,23]],"date-time":"2018-08-23T00:00:00Z","timestamp":1534982400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cram\u00e9r\u2013Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cram\u00e9r\u2013Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.<\/jats:p>","DOI":"10.3390\/e20090628","type":"journal-article","created":{"date-parts":[[2018,8,24]],"date-time":"2018-08-24T03:42:31Z","timestamp":1535082151000},"page":"628","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":60,"title":["Frequentist and Bayesian Quantum Phase Estimation"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0628-6153","authenticated-orcid":false,"given":"Yan","family":"Li","sequence":"first","affiliation":[{"name":"Institute of Theoretical Physics and Department of Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China"}]},{"given":"Luca","family":"Pezz\u00e8","sequence":"additional","affiliation":[{"name":"QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4203-0366","authenticated-orcid":false,"given":"Manuel","family":"Gessner","sequence":"additional","affiliation":[{"name":"QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy"}]},{"given":"Zhihong","family":"Ren","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics and Department of Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China"}]},{"given":"Weidong","family":"Li","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics and Department of Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China"}]},{"given":"Augusto","family":"Smerzi","sequence":"additional","affiliation":[{"name":"QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,23]]},"reference":[{"key":"ref_1","first-page":"275","article-title":"Ein neuer Interferenzrefraktor","volume":"11","author":"Zehnder","year":"1891","journal-title":"Zeitschrift f\u00fcr Instrumentenkunde"},{"key":"ref_2","first-page":"89","article-title":"Ueber einen Interferenzrefraktor","volume":"12","author":"Mach","year":"1892","journal-title":"Zeitschrift f\u00fcr Instrumentenkunde"},{"key":"ref_3","unstructured":"Ramsey, N.F. 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