{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:22:02Z","timestamp":1760239322264,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,10,26]],"date-time":"2020-10-26T00:00:00Z","timestamp":1603670400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 11775084"],"award-info":[{"award-number":["No. 11775084"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and R\u00e9nyi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in cases where the system density matrix obeys the Gorini\u2013Kossakowski\u2013Lindblad\u2013Sudarshan equation.<\/jats:p>","DOI":"10.3390\/e22111219","type":"journal-article","created":{"date-parts":[[2020,10,27]],"date-time":"2020-10-27T09:22:45Z","timestamp":1603790565000},"page":"1219","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Quantum Weak Invariants: Dynamical Evolution of Fluctuations and Correlations"],"prefix":"10.3390","volume":"22","author":[{"given":"Zeyi","family":"Shi","sequence":"first","affiliation":[{"name":"Department of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sumiyoshi","family":"Abe","sequence":"additional","affiliation":[{"name":"Department of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China"},{"name":"Institute of Physics, Kazan Federal University, Kazan 420008, Russia"},{"name":"Department of Natural and Mathematical Sciences, Turin Polytechnic University in Tashkent, Tashkent 100095, Uzbekistan"},{"name":"ESIEA, 9 Rue Vesale, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1063\/1.881363","article-title":"Boltzmann\u2019s entropy and time\u2019s arrow","volume":"46","author":"Lebowitz","year":"1993","journal-title":"Physics Today"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"034002","DOI":"10.1088\/1402-4896\/aaece6","article-title":"The direction of time and Boltzmann\u2019s time hypothesis","volume":"94","author":"Klimenko","year":"2019","journal-title":"Phys. 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