{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:52QEE4KLMLNIDDHIGWW5RRSDRG","short_pith_number":"pith:52QEE4KL","canonical_record":{"source":{"id":"1602.00233","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-01-31T11:51:34Z","cross_cats_sorted":["cs.IT","math.IT","math.MP","math.PR","quant-ph"],"title_canon_sha256":"7650abe706b7b88d0b7918ca5e4cc3b7908ebf110f4f3305dc941f6c447d70ff","abstract_canon_sha256":"b7ec3249d2ff89d6d8dff2bb8b1f8ca07c3d7059535272770bac3037fa66230f"},"schema_version":"1.0"},"canonical_sha256":"eea042714b62da818ce835add8c64389890a5fae244c3848069b888f26b6eb73","source":{"kind":"arxiv","id":"1602.00233","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.00233","created_at":"2026-05-18T01:07:58Z"},{"alias_kind":"arxiv_version","alias_value":"1602.00233v1","created_at":"2026-05-18T01:07:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00233","created_at":"2026-05-18T01:07:58Z"},{"alias_kind":"pith_short_12","alias_value":"52QEE4KLMLNI","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"52QEE4KLMLNIDDHI","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"52QEE4KL","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:52QEE4KLMLNIDDHIGWW5RRSDRG","target":"record","payload":{"canonical_record":{"source":{"id":"1602.00233","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-01-31T11:51:34Z","cross_cats_sorted":["cs.IT","math.IT","math.MP","math.PR","quant-ph"],"title_canon_sha256":"7650abe706b7b88d0b7918ca5e4cc3b7908ebf110f4f3305dc941f6c447d70ff","abstract_canon_sha256":"b7ec3249d2ff89d6d8dff2bb8b1f8ca07c3d7059535272770bac3037fa66230f"},"schema_version":"1.0"},"canonical_sha256":"eea042714b62da818ce835add8c64389890a5fae244c3848069b888f26b6eb73","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:58.317273Z","signature_b64":"j2d87f3jf8iZ83u3nvpotrAp6yUAWVs+mg89RiwPfrhhlbKUz2gndxyJot3VlaK4EP1rIa17gntnct1T3p6dCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eea042714b62da818ce835add8c64389890a5fae244c3848069b888f26b6eb73","last_reissued_at":"2026-05-18T01:07:58.316820Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:58.316820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.00233","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:07:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TClzq+lUgRS3Siov9jK+DcImK6EisFbKqNd8RYfaC2DD/mfPAH5DiRZEkbX14WFn2LHjuYUUk0Rjy8aArUpABw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T12:32:25.311479Z"},"content_sha256":"374a1ed064df453cc72212423633eb34d8d59bf7732c0bfffff79636b34441af","schema_version":"1.0","event_id":"sha256:374a1ed064df453cc72212423633eb34d8d59bf7732c0bfffff79636b34441af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:52QEE4KLMLNIDDHIGWW5RRSDRG","target":"graph","payload":{"graph_snapshot":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:07:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vg+8OHsk2WPD+X1lru5xGdSqOLl0AqhEQjUj9qKRsWQOQQCUh85k/N31D/nNAeHNq+5IpMtG6kJzRTzNde58BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T12:32:25.311852Z"},"content_sha256":"3e59b02138566a536cde225887d65c1702538925acc970d8f3fd06620cd38050","schema_version":"1.0","event_id":"sha256:3e59b02138566a536cde225887d65c1702538925acc970d8f3fd06620cd38050"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/52QEE4KLMLNIDDHIGWW5RRSDRG/bundle.json","state_url":"https://pith.science/pith/52QEE4KLMLNIDDHIGWW5RRSDRG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/52QEE4KLMLNIDDHIGWW5RRSDRG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-11T12:32:25Z","links":{"resolver":"https://pith.science/pith/52QEE4KLMLNIDDHIGWW5RRSDRG","bundle":"https://pith.science/pith/52QEE4KLMLNIDDHIGWW5RRSDRG/bundle.json","state":"https://pith.science/pith/52QEE4KLMLNIDDHIGWW5RRSDRG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/52QEE4KLMLNIDDHIGWW5RRSDRG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:52QEE4KLMLNIDDHIGWW5RRSDRG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b7ec3249d2ff89d6d8dff2bb8b1f8ca07c3d7059535272770bac3037fa66230f","cross_cats_sorted":["cs.IT","math.IT","math.MP","math.PR","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-01-31T11:51:34Z","title_canon_sha256":"7650abe706b7b88d0b7918ca5e4cc3b7908ebf110f4f3305dc941f6c447d70ff"},"schema_version":"1.0","source":{"id":"1602.00233","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.00233","created_at":"2026-05-18T01:07:58Z"},{"alias_kind":"arxiv_version","alias_value":"1602.00233v1","created_at":"2026-05-18T01:07:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00233","created_at":"2026-05-18T01:07:58Z"},{"alias_kind":"pith_short_12","alias_value":"52QEE4KLMLNI","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"52QEE4KLMLNIDDHI","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"52QEE4KL","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:3e59b02138566a536cde225887d65c1702538925acc970d8f3fd06620cd38050","target":"graph","created_at":"2026-05-18T01:07:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Hao-Chung Cheng, Min-Hsiu Hsieh","cross_cats":["cs.IT","math.IT","math.MP","math.PR","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-01-31T11:51:34Z","title":"Characterisations of Matrix and Operator-Valued $\\Phi$-Entropies, and Operator Efron-Stein Inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00233","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:374a1ed064df453cc72212423633eb34d8d59bf7732c0bfffff79636b34441af","target":"record","created_at":"2026-05-18T01:07:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b7ec3249d2ff89d6d8dff2bb8b1f8ca07c3d7059535272770bac3037fa66230f","cross_cats_sorted":["cs.IT","math.IT","math.MP","math.PR","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-01-31T11:51:34Z","title_canon_sha256":"7650abe706b7b88d0b7918ca5e4cc3b7908ebf110f4f3305dc941f6c447d70ff"},"schema_version":"1.0","source":{"id":"1602.00233","kind":"arxiv","version":1}},"canonical_sha256":"eea042714b62da818ce835add8c64389890a5fae244c3848069b888f26b6eb73","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eea042714b62da818ce835add8c64389890a5fae244c3848069b888f26b6eb73","first_computed_at":"2026-05-18T01:07:58.316820Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:58.316820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"j2d87f3jf8iZ83u3nvpotrAp6yUAWVs+mg89RiwPfrhhlbKUz2gndxyJot3VlaK4EP1rIa17gntnct1T3p6dCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:58.317273Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.00233","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:374a1ed064df453cc72212423633eb34d8d59bf7732c0bfffff79636b34441af","sha256:3e59b02138566a536cde225887d65c1702538925acc970d8f3fd06620cd38050"],"state_sha256":"467bb87cd43667a912e76d1879cb8de4b819b44e384f4c4472483e9c6e0fcc7a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yNJERI6aoUPF4Ih2ig/oIpDm4bW9FMC5iP6GuWnz6RH59nH99mJ7ZKW81Zr4xqG475S3AHTRdCrnkVD9CvFDDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T12:32:25.317326Z","bundle_sha256":"ec38e3b90d8ac9a471071e16f38acfeedd4e3d1e3bb867ab2af2e0c36183d594"}}