{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/170602212","title":"Maximum and minimum entropy states yielding local continuity bounds","arxiv_id":"1706.02212","date":"2017-06-07","proceeding":null,"authors":["Eric P. Hanson","Nilanjana Datta"],"abstract":"Given an arbitrary quantum state ($\\sigma$), we obtain an explicit construction of a state $\\rho^*_\\varepsilon(\\sigma)$ (resp. $\\rho_{*,\\varepsilon}(\\sigma)$) which has the maximum (resp. minimum) entropy among all states which lie in a specified neighbourhood ($\\varepsilon$-ball) of $\\sigma$. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of $\\sigma$. The states $\\rho^*_\\varepsilon(\\sigma)$ and $\\rho_{*,\\varepsilon}(\\sigma)$ depend only on the geometry of the $\\varepsilon$-ball and are in fact optimizers for a larger class of entropies. These include the R\\'enyi entropy and the min- and max- entropies. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely a necessary and sufficient condition under which a state maximizes a concave and G\\^ateaux-differentiable function in an $\\varepsilon$-ball around a given state $\\sigma$. Examples of such a function include the von Neumann entropy, and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat's Rule, and majorization theory.","url_abs":"http://arxiv.org/abs/1706.02212v2","url_pdf":"http://arxiv.org/pdf/1706.02212v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"170602212","repo_url":"https://github.com/ericphanson/MajorizationExtrema.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}