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Quantum correlated states, such as squeezed states, enable phase estimation beyond the shot-noise limit, and in principle approach the ultimate quantum limit in precision, when paired with optimal quantum measurements. However, physical realizations of optimal quantum measurements for optical phase estimation with quantum-correlated states are still unknown. Here we address this problem by introducing an adaptive Gaussian measurement strategy for optical phase estimation with squeezed vacuum states that, by construction, approaches the quantum limit in precision. This strategy builds from a comprehensive set of locally optimal POVMs through rotations and homodyne measurements and uses the Adaptive Quantum State Estimation framework for optimizing the adaptive measurement process, which, under certain regularity conditions, guarantees asymptotic optimality for this quantum parameter estimation problem. As a result, the adaptive phase estimation strategy based on locally-optimal homodyne measurements achieves the quantum limit within the phase interval of [<\/mml:mo>0<\/mml:mn>,<\/mml:mo>&#x03C0;<\/mml:mi>\/<\/mml:mo><\/mml:mrow>2<\/mml:mn>)<\/mml:mo><\/mml:math>. Furthermore, we generalize this strategy by including heterodyne measurements, enabling phase estimation across the full range of phases from [<\/mml:mo>0<\/mml:mn>,<\/mml:mo>&#x03C0;<\/mml:mi>)<\/mml:mo><\/mml:math>, where squeezed vacuum allows for unambiguous phase encoding. Remarkably, for this phase interval, which is the maximum range of phases that can be encoded in squeezed vacuum, this estimation strategy maintains an asymptotic quantum-optimal performance, representing a significant advancement in quantum metrology.<\/jats:p>","DOI":"10.22331\/q-2024-09-25-1480","type":"journal-article","created":{"date-parts":[[2024,9,25]],"date-time":"2024-09-25T12:40:31Z","timestamp":1727268031000},"page":"1480","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Adaptive Phase Estimation with Squeezed Vacuum Approaching the Quantum Limit"],"prefix":"10.22331","volume":"8","author":[{"given":"M. A.","family":"Rodr\u00edguez-Garc\u00eda","sequence":"first","affiliation":[{"name":"Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA"}]},{"given":"F. 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(2005).","DOI":"10.1142\/5630"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Holevo, A. ``Probabilistic and statistical aspects of quantum theory''. Edizioni della Normale. (2011).","DOI":"10.1007\/978-88-7642-378-9"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Aasi, J., Abadie, J., and Abbott, t. ``Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light''. Nature Photon 7, 613\u2013619 (2013).","DOI":"10.1038\/nphoton.2013.177"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Gatti, A., Brambilla, E., and Lugiato, L. ``Chapter 5 Quantum imaging''. Page 251\u2013348. Elsevier. (2008).","DOI":"10.1016\/s0079-6638(07)51005-x"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Kruse, I., Lange, K., Peise, J., L\u00fccke, B., Pezz\u00e8, L., Arlt, J., Ertmer, W., Lisdat, C., Santos, L., Smerzi, A., and Klempt, C. ``Improvement of an atomic clock using squeezed vacuum''. Phys. Rev. Lett. 117, 143004 (2016).","DOI":"10.1103\/PhysRevLett.117.143004"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Danilin, S., Lebedev, A. V., Veps\u00e4l\u00e4inen, A., Lesovik, G. B., Blatter, G., and Paraoanu, G. S. ``Quantum-enhanced magnetometry by phase estimation algorithms with a single artificial atom''. npj Quantum Inf 4 (2018).","DOI":"10.1038\/s41534-018-0078-y"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Gilchrist, A., Nemoto, K., Munro, W. J., Ralph, T. C., Glancy, S., Braunstein, S. L., and Milburn, G. J. ``Schr\u00f6dinger cats and their power for quantum information processing''. J. Opt. B: Quantum Semiclass. Opt. 6, S828\u2013S833 (2004).","DOI":"10.1088\/1464-4266\/6\/8\/032"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Wiseman, H. M. and Milburn, G. J. ``Quantum measurement and control''. Cambridge University Press. 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A 73, 033821 (2006).","DOI":"10.1103\/PhysRevA.73.033821"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Fujiwara, A. ``Strong consistency and asymptotic efficiency for adaptive quantum estimation problems''. J. Phys. A: Math. Gen. 39, 12489\u201312504 (2006).","DOI":"10.1088\/0305-4470\/39\/40\/014"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Nagaoka, H. ``On Fisher information of quantum statistical models''. Page 113\u2013124. WORLD SCIENTIFIC. (2005).","DOI":"10.1142\/9789812563071_0010"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Chiribella, G. and Mauro D\u2019Ariano, G. ``Extremal covariant measurements''. J. Math. Phys. 47 (2006).","DOI":"10.1063\/1.2349481"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Beneduci, R. ``On the relationships between the moments of a POVM and the generator of the von Neumann algebra it generates''. 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