{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T00:51:33Z","timestamp":1775263893755,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2016,9,1]],"date-time":"2016-09-01T00:00:00Z","timestamp":1472688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Motivated by the importance of contextuality and a work on the robustness of the entanglement of mixed quantum states, the robustness of contextuality (RoC)      R C   ( e )      of an empirical model e against non-contextual noises was introduced and discussed in Science China Physics, Mechanics and Astronomy (59(4) and 59(9), 2016). Because noises are not always non-contextual, this paper introduces and discusses the generalized robustness of contextuality (GRoC)      R g   ( e )      of an empirical model e against general noises. It is proven that      R g   ( e )  = 0     if and only if e is non-contextual. This means that the quantity     R g     can be used to distinguish contextual empirical models from non-contextual ones. It is also shown that the function     R g     is convex on the set of all empirical models and continuous on the set of all no-signaling empirical models. For any two empirical models e and f such that the generalized relative robustness of e with respect to f is finite, a fascinating relationship between the GRoCs of e and f is proven, which reads      R g   ( e )   R g   ( f )  \u2264 1    . Lastly, for any n-cycle contextual box e, a relationship between the GRoC      R g   ( e )      and the extent     \u0394 e     of violating the non-contextual inequalities is established.<\/jats:p>","DOI":"10.3390\/e18090297","type":"journal-article","created":{"date-parts":[[2016,9,1]],"date-time":"2016-09-01T10:04:28Z","timestamp":1472724268000},"page":"297","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Generalized Robustness of Contextuality"],"prefix":"10.3390","volume":"18","author":[{"given":"Huixian","family":"Meng","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2141-825X","authenticated-orcid":false,"given":"Huaixin","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenhua","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Ethnic Nationalities Education, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yajing","family":"Fan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liang","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,9,1]]},"reference":[{"key":"ref_1","first-page":"59","article-title":"The Problem of Hidden Variables in Quantum Mechanics","volume":"17","author":"Kochen","year":"1967","journal-title":"J. Math. Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4089","DOI":"10.1103\/PhysRevA.55.4089","article-title":"Quantum code words contradict local realism","volume":"55","author":"DiVincenzo","year":"1997","journal-title":"Phys. Rev. A"},{"key":"ref_3","unstructured":"Galv\u00e3o, E.F. (2002). Foundations of quantum theory and quantum information applications. [Ph.D. Thesis, Oxford University]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"012325","DOI":"10.1103\/PhysRevA.72.012325","article-title":"Kochen-Specker theorem as a precondition for secure quantum key distribution","volume":"72","author":"Nagata","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"052310","DOI":"10.1103\/PhysRevA.77.052310","article-title":"Quantum advantages in classically defined tasks","volume":"77","author":"Aharon","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1038\/nature13460","article-title":"Contextuality supplies the \u201cmagic\u201d for quantum computation","volume":"510","author":"Howard","year":"2014","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1204011","DOI":"10.1103\/PhysRevLett.112.120401","article-title":"Quantifying contextuality","volume":"112","author":"Grudka","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1103\/PhysRev.47.777","article-title":"Can quantum-mechanical description of physical reality be considered complete?","volume":"47","author":"Einstein","year":"1935","journal-title":"Phys. Rev."},{"key":"ref_9","unstructured":"Bell, J. (1987). Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge University Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"113036","DOI":"10.1088\/1367-2630\/13\/11\/113036","article-title":"The Sheaf-theoretic structure of non-locality and contextuality","volume":"13","author":"Abramsky","year":"2011","journal-title":"New J. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1103\/PhysRevA.59.141","article-title":"Robustness of entanglement","volume":"59","author":"Vidal","year":"1999","journal-title":"Phys. Rev. A."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"640303","DOI":"10.1007\/s11433-016-5795-7","article-title":"The robustness of contextuality and the contextuality cost of empirical models","volume":"59","author":"Meng","year":"2016","journal-title":"Sci. China Phys. Mech. Astron."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"100311","DOI":"10.1007\/s11433-016-0210-2","article-title":"Continuity of the robustness of contextuality of empirical models","volume":"59","author":"Meng","year":"2016","journal-title":"Sci. China Phys. Mech. Astron."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"054305","DOI":"10.1103\/PhysRevA.67.054305","article-title":"Generalized robustness of entanglement","volume":"67","author":"Steiner","year":"2003","journal-title":"Phys. Rev. A."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"062114","DOI":"10.1103\/PhysRevA.85.062114","article-title":"Logical Bell inequalities","volume":"85","author":"Abramsky","year":"2012","journal-title":"Phys. Rev. A."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"022118","DOI":"10.1103\/PhysRevA.88.022118","article-title":"All non-contextuality inequalities for the n-cycle scenario","volume":"88","author":"Quintino","year":"2013","journal-title":"Phys. Rev. A."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","article-title":"Proposed experiment to test Local Hidden-Variable theories","volume":"23","author":"Clauser","year":"1969","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"210401","DOI":"10.1103\/PhysRevLett.101.210401","article-title":"Experimentally testable state-independent quantum contextuality","volume":"101","author":"Cabello","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1038\/nature08172","article-title":"State-independent experimental test of quantum contextuality","volume":"460","author":"Kirchmair","year":"2009","journal-title":"Nature"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"160405","DOI":"10.1103\/PhysRevLett.103.160405","article-title":"State-independent quantum contextuality with single photons","volume":"103","author":"Amselem","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2706","DOI":"10.1038\/srep02706","article-title":"An experimental proposal for revealing contextuality in almost all qutrit states","volume":"3","author":"Lapkiewicz","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (1985). Matrix Analysis, Cambridge Univerisity Press.","DOI":"10.1017\/CBO9780511810817"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/9\/297\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:29:47Z","timestamp":1760210987000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/9\/297"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,1]]},"references-count":22,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2016,9]]}},"alternative-id":["e18090297"],"URL":"https:\/\/doi.org\/10.3390\/e18090297","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,9,1]]}}}