{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T10:23:01Z","timestamp":1775038981428,"version":"3.50.1"},"reference-count":57,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2010,7,20]],"date-time":"2010-07-20T00:00:00Z","timestamp":1279584000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Convex roof extensions are widely used to create entanglement measures in quantum information theory. The aim of the article is to present some tools which could be helpful for their treatment. Sections 2 and 3 introduce into the subject. It follows descriptions of the Wootters' method, of the \"subtraction procedure\", and examples on how to use symmetries.<\/jats:p>","DOI":"10.3390\/e12071799","type":"journal-article","created":{"date-parts":[[2010,7,21]],"date-time":"2010-07-21T02:19:50Z","timestamp":1279678790000},"page":"1799-1832","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":64,"title":["Roofs and Convexity"],"prefix":"10.3390","volume":"12","author":[{"given":"Armin","family":"Uhlmann","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, University of Leipzig, PB 100 920, D-04009 Leipzig, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2010,7,20]]},"reference":[{"key":"ref_1","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). 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A 3-dimensional null-space forces CT to be constant."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"032310","DOI":"10.1103\/PhysRevA.77.032310","article-title":"Tangles of superpositions and the convex-roof extension","volume":"77","author":"Osterloh","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"0623307","DOI":"10.1103\/PhysRevA.64.062307","article-title":"Entanglement measures under symmetry","volume":"64","author":"Vollbrecht","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2625","DOI":"10.1103\/PhysRevLett.85.2625","article-title":"The entanglement of formation for isotropic states","volume":"85","author":"Terhal","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Ohya, M., and Petz, D. (1993). Quantum Entropy and Its Use, Springer.","DOI":"10.1007\/978-3-642-57997-4"},{"key":"ref_29","unstructured":"A flat roof point can allow for several optimal decompositions, some of them flat and some of them not flat."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/BF02228857","article-title":"Entropy defect and information for two quantum states","volume":"2","author":"Levitan","year":"1994","journal-title":"Open Sys. Inf. Dyn."},{"key":"ref_31","unstructured":"Thirring, W. 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