{"paper":{"title":"Characterisations of Matrix and Operator-Valued $\\Phi$-Entropies, and Operator Efron-Stein Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","math.MP","math.PR","quant-ph"],"primary_cat":"math-ph","authors_text":"Hao-Chung Cheng, Min-Hsiu Hsieh","submitted_at":"2016-01-31T11:51:34Z","abstract_excerpt":"We derive new characterisations of the matrix $\\mathrm{\\Phi}$-entropy functionals introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014]. Notably, all known equivalent characterisations of the classical $\\Phi$-entropies have their matrix correspondences. Next, we propose an operator-valued generalisation of the matrix $\\Phi$-entropy functionals, and prove their subadditivity under L\\\"owner partial ordering. Our results demonstrate that the subadditivity of operator-valued $\\Phi$-entropies is equivalent to the convexity of various related functions. This result can be used to demonstrate an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00233","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}