{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T03:59:23Z","timestamp":1780631963426,"version":"3.54.1"},"reference-count":41,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,16]],"date-time":"2021-03-16T00:00:00Z","timestamp":1615852800000},"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>Weak measurements have been under intensive investigation in both experiment and theory. Numerous experiments have indicated that the amplified meter shift is produced by the post-selection, yielding an improved precision compared to conventional methods. However, this amplification effect comes at the cost of a reduced rate of acquiring data, which leads to an increasing uncertainty to determine the level of meter shift. From this point of view, a number of theoretical works have suggested that weak measurements cannot improve the precision, or even damage the metrology information due to the post-selection. In this review, we give a comprehensive analysis of the weak measurements to justify their positive effect on prompting measurement precision. As a further step, we introduce two modified weak measurement protocols to boost the precision beyond the standard quantum limit. Compared to previous works beating the standard quantum limit, these protocols are free of using entangled or squeezed states. The achieved precision outperforms that of the conventional method by two orders of magnitude and attains a practical Heisenberg scaling up to n=106 photons.<\/jats:p>","DOI":"10.3390\/e23030354","type":"journal-article","created":{"date-parts":[[2021,3,16]],"date-time":"2021-03-16T12:01:02Z","timestamp":1615896062000},"page":"354","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Beating Standard Quantum Limit with Weak Measurement"],"prefix":"10.3390","volume":"23","author":[{"given":"Geng","family":"Chen","sequence":"first","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peng","family":"Yin","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wen-Hao","family":"Zhang","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gong-Chu","family":"Li","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chuan-Feng","family":"Li","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guang-Can","family":"Guo","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1103\/PhysRevLett.60.1351","article-title":"How the result of a measurement of a component of the spin of a spin-1\/2 particle can turn out to be 100","volume":"10","author":"Aharonov","year":"1988","journal-title":"Phys. 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