{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:13:43Z","timestamp":1760166823375,"version":"build-2065373602"},"reference-count":72,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T00:00:00Z","timestamp":1626134400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMR-2010127"],"award-info":[{"award-number":["DMR-2010127"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009566","name":"Foundational Questions Institute","doi-asserted-by":"publisher","award":["FQXi-RFP-1808"],"award-info":[{"award-number":["FQXi-RFP-1808"]}],"id":[{"id":"10.13039\/100009566","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In classical thermodynamics the Euler relation is an expression for the internal energy as a sum of the products of canonical pairs of extensive and intensive variables. For quantum systems the situation is more intricate, since one has to account for the effects of the measurement back action. To this end, we derive a quantum analog of the Euler relation, which is governed by the information retrieved by local quantum measurements. The validity of the relation is demonstrated for the collective dissipation model, where we find that thermodynamic behavior is exhibited in the weak-coupling regime.<\/jats:p>","DOI":"10.3390\/e23070889","type":"journal-article","created":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T22:25:31Z","timestamp":1626215131000},"page":"889","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quantum Euler Relation for Local Measurements"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2405-7883","authenticated-orcid":false,"given":"Akram","family":"Touil","sequence":"first","affiliation":[{"name":"Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2285-3840","authenticated-orcid":false,"given":"Kevin","family":"Weber","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA"},{"name":"Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0504-6932","authenticated-orcid":false,"given":"Sebastian","family":"Deffner","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA"},{"name":"Instituto de F\u00edsica \u2018Gleb Wataghin\u2019, Universidade Estadual de Campinas, Campinas 13083-859, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,13]]},"reference":[{"key":"ref_1","unstructured":"Callen, H.B. 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