{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T00:13:46Z","timestamp":1780359226414,"version":"3.54.1"},"reference-count":86,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T00:00:00Z","timestamp":1600905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100010434","name":"\u201cla Caixa\u201d Foundation","doi-asserted-by":"publisher","award":["ID 100010434, fellowship code LCF\/BQ\/DI19\/11730023"],"award-info":[{"award-number":["ID 100010434, fellowship code LCF\/BQ\/DI19\/11730023"]}],"id":[{"id":"10.13039\/100010434","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001711","name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","doi-asserted-by":"publisher","award":["Ambizione PZ00P2-186067"],"award-info":[{"award-number":["Ambizione PZ00P2-186067"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"publisher","award":["Doctoral Prize"],"award-info":[{"award-number":["Doctoral Prize"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Marie Sklodowska-Curie","award":["grant agreement 239 No 713729"],"award-info":[{"award-number":["grant agreement 239 No 713729"]}]},{"DOI":"10.13039\/501100010198","name":"MINECO","doi-asserted-by":"publisher","award":["QIBEQI FIS2016-80773-P, 240 Severo Ochoa SEV-2015-0522"],"award-info":[{"award-number":["QIBEQI FIS2016-80773-P, 240 Severo Ochoa SEV-2015-0522"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002809","name":"Generalitat de Catalunya","doi-asserted-by":"publisher","award":["SGR 1381 and CERCA Programme"],"award-info":[{"award-number":["SGR 1381 and CERCA Programme"]}],"id":[{"id":"10.13039\/501100002809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We review and connect different frameworks where it emerges in the quantum regime: adiabatically driven closed systems, time-dependent Lindblad master equations, and discrete processes. A geometric lower bound on entropy production in finite-time is then presented, which represents a quantum generalisation of the original classical bound. Following this, we review and develop some general principles for the optimisation of thermodynamic processes in the linear-response regime. These include constant speed of control variation according to the thermodynamic metric, absence of quantum coherence, and optimality of small cycles around the point of maximal ratio between heat capacity and relaxation time for Carnot engines.<\/jats:p>","DOI":"10.3390\/e22101076","type":"journal-article","created":{"date-parts":[[2020,9,25]],"date-time":"2020-09-25T01:39:33Z","timestamp":1600997973000},"page":"1076","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":82,"title":["Geometric Optimisation of Quantum Thermodynamic Processes"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5019-7823","authenticated-orcid":false,"given":"Paolo","family":"Abiuso","sequence":"first","affiliation":[{"name":"ICFO\u2014Institut de Ci\u00e8ncies Fot\u00f2niques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6275-0446","authenticated-orcid":false,"given":"Harry J. D.","family":"Miller","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4658-0632","authenticated-orcid":false,"given":"Mart\u00ed","family":"Perarnau-Llobet","sequence":"additional","affiliation":[{"name":"D\u00e9partement de Physique Appliqu\u00e9e, Universit\u00e9 de Gen\u00e8ve, 1206 Gen\u00e8ve, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6395-3971","authenticated-orcid":false,"given":"Matteo","family":"Scandi","sequence":"additional","affiliation":[{"name":"ICFO\u2014Institut de Ci\u00e8ncies Fot\u00f2niques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,24]]},"reference":[{"key":"ref_1","unstructured":"Landau, L.D., and Lifshitz, E.M. 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