{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:4DUDRAMABTCDL2HDBSS747VCX4","short_pith_number":"pith:4DUDRAMA","canonical_record":{"source":{"id":"1908.08942","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-22T23:14:50Z","cross_cats_sorted":["math-ph","math.MP","math.PR","quant-ph"],"title_canon_sha256":"9547182a6bd695338fdb7d36aaa7ad1417147039ed15bdb89e634b2ab9c9651b","abstract_canon_sha256":"13246d76d83dc6bc23ea24a114dcd960a1f9374d8a638fd7405e351fec335766"},"schema_version":"1.0"},"canonical_sha256":"e0e83881800cc435e8e30ca5fe7ea2bf2f98bba0d43e697fb2b050f19c1b9497","source":{"kind":"arxiv","id":"1908.08942","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08942","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08942v3","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08942","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"pith_short_12","alias_value":"4DUDRAMABTCD","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"pith_short_16","alias_value":"4DUDRAMABTCDL2HD","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"pith_short_8","alias_value":"4DUDRAMA","created_at":"2026-07-05T03:14:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:4DUDRAMABTCDL2HDBSS747VCX4","target":"record","payload":{"canonical_record":{"source":{"id":"1908.08942","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-22T23:14:50Z","cross_cats_sorted":["math-ph","math.MP","math.PR","quant-ph"],"title_canon_sha256":"9547182a6bd695338fdb7d36aaa7ad1417147039ed15bdb89e634b2ab9c9651b","abstract_canon_sha256":"13246d76d83dc6bc23ea24a114dcd960a1f9374d8a638fd7405e351fec335766"},"schema_version":"1.0"},"canonical_sha256":"e0e83881800cc435e8e30ca5fe7ea2bf2f98bba0d43e697fb2b050f19c1b9497","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:14:31.526341Z","signature_b64":"Sr8NsKQt3h0tYc5geiZiZSgc8RnTdmvd6U7Ym0duicrQt+np6iVyHKRLvCZG3p2qREo2BSPuI2MmYZJc5McSCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0e83881800cc435e8e30ca5fe7ea2bf2f98bba0d43e697fb2b050f19c1b9497","last_reissued_at":"2026-07-05T03:14:31.525841Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:14:31.525841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.08942","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:14:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"stcTBsebbpvg0vg5yLh7ORjG8ivo4F91RkLX1McLuWxYDTf3JAUUgyRCg796/+4CtRyhsn2OXC3O2npKxMkEAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T18:45:55.936803Z"},"content_sha256":"0b236776617893ae1a87c55aee0ae0b302d38b44f0903110dce4956151f080cf","schema_version":"1.0","event_id":"sha256:0b236776617893ae1a87c55aee0ae0b302d38b44f0903110dce4956151f080cf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:4DUDRAMABTCDL2HDBSS747VCX4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lyapunov exponents for Quantum Channels: an entropy formula and generic properties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR","quant-ph"],"primary_cat":"math.DS","authors_text":"Artur O. Lopes, Jader E. Brasil, Josue Knorst","submitted_at":"2019-08-22T23:14:50Z","abstract_excerpt":"We denote by $M_k$ the set of $k$ by $k$ matrices with complex entries. We consider quantum channels $\\phi_L$ of the form: given a measurable function $L:M_k\\to M_k$ and a measure $\\mu$ on $M_k$ we define the linear operator $\\phi_L:M_k \\to M_k$, by the law $\\rho \\,\\to\\,\\phi_L(\\rho) = \\int_{M_k} L(v) \\rho L(v)^\\dagger \\, \\dm(v).$\n  On a previous work the authors show that for a fixed measure $\\mu$ it is generic on the function $L$ the $\\Phi$-Erg property (also irreducibility). Here we will show that the purification property is also generic on $L$ for a fixed $\\mu$.\n  Given $L$ and $\\mu$ there"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08942","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.08942/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T03:14:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kZgrMwkD5ovAxGnFY8IJvF9IcW2WyNkN0ut48+8aUVVujFCufZ51Aq2iLSmvP3Au54QCiYIJMWEigNLo6ZQHBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T18:45:55.937317Z"},"content_sha256":"35c239e47f4c3ce869a2dd74c1fd9877d4853c62e9c097da326cde63026296c5","schema_version":"1.0","event_id":"sha256:35c239e47f4c3ce869a2dd74c1fd9877d4853c62e9c097da326cde63026296c5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4DUDRAMABTCDL2HDBSS747VCX4/bundle.json","state_url":"https://pith.science/pith/4DUDRAMABTCDL2HDBSS747VCX4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4DUDRAMABTCDL2HDBSS747VCX4/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":"2026-08-16T18:45:55Z","links":{"resolver":"https://pith.science/pith/4DUDRAMABTCDL2HDBSS747VCX4","bundle":"https://pith.science/pith/4DUDRAMABTCDL2HDBSS747VCX4/bundle.json","state":"https://pith.science/pith/4DUDRAMABTCDL2HDBSS747VCX4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4DUDRAMABTCDL2HDBSS747VCX4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:4DUDRAMABTCDL2HDBSS747VCX4","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":"13246d76d83dc6bc23ea24a114dcd960a1f9374d8a638fd7405e351fec335766","cross_cats_sorted":["math-ph","math.MP","math.PR","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-22T23:14:50Z","title_canon_sha256":"9547182a6bd695338fdb7d36aaa7ad1417147039ed15bdb89e634b2ab9c9651b"},"schema_version":"1.0","source":{"id":"1908.08942","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08942","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08942v3","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08942","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"pith_short_12","alias_value":"4DUDRAMABTCD","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"pith_short_16","alias_value":"4DUDRAMABTCDL2HD","created_at":"2026-07-05T03:14:31Z"},{"alias_kind":"pith_short_8","alias_value":"4DUDRAMA","created_at":"2026-07-05T03:14:31Z"}],"graph_snapshots":[{"event_id":"sha256:35c239e47f4c3ce869a2dd74c1fd9877d4853c62e9c097da326cde63026296c5","target":"graph","created_at":"2026-07-05T03:14:31Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.08942/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We denote by $M_k$ the set of $k$ by $k$ matrices with complex entries. We consider quantum channels $\\phi_L$ of the form: given a measurable function $L:M_k\\to M_k$ and a measure $\\mu$ on $M_k$ we define the linear operator $\\phi_L:M_k \\to M_k$, by the law $\\rho \\,\\to\\,\\phi_L(\\rho) = \\int_{M_k} L(v) \\rho L(v)^\\dagger \\, \\dm(v).$\n  On a previous work the authors show that for a fixed measure $\\mu$ it is generic on the function $L$ the $\\Phi$-Erg property (also irreducibility). Here we will show that the purification property is also generic on $L$ for a fixed $\\mu$.\n  Given $L$ and $\\mu$ there","authors_text":"Artur O. Lopes, Jader E. Brasil, Josue Knorst","cross_cats":["math-ph","math.MP","math.PR","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-22T23:14:50Z","title":"Lyapunov exponents for Quantum Channels: an entropy formula and generic properties"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08942","kind":"arxiv","version":3},"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":"sha256:0b236776617893ae1a87c55aee0ae0b302d38b44f0903110dce4956151f080cf","target":"record","created_at":"2026-07-05T03:14:31Z","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":"13246d76d83dc6bc23ea24a114dcd960a1f9374d8a638fd7405e351fec335766","cross_cats_sorted":["math-ph","math.MP","math.PR","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-08-22T23:14:50Z","title_canon_sha256":"9547182a6bd695338fdb7d36aaa7ad1417147039ed15bdb89e634b2ab9c9651b"},"schema_version":"1.0","source":{"id":"1908.08942","kind":"arxiv","version":3}},"canonical_sha256":"e0e83881800cc435e8e30ca5fe7ea2bf2f98bba0d43e697fb2b050f19c1b9497","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e0e83881800cc435e8e30ca5fe7ea2bf2f98bba0d43e697fb2b050f19c1b9497","first_computed_at":"2026-07-05T03:14:31.525841Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:14:31.525841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Sr8NsKQt3h0tYc5geiZiZSgc8RnTdmvd6U7Ym0duicrQt+np6iVyHKRLvCZG3p2qREo2BSPuI2MmYZJc5McSCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:14:31.526341Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.08942","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b236776617893ae1a87c55aee0ae0b302d38b44f0903110dce4956151f080cf","sha256:35c239e47f4c3ce869a2dd74c1fd9877d4853c62e9c097da326cde63026296c5"],"state_sha256":"a355d2ee4c889de1421a008eaf77389e6604b689b60785b6521c79350c6b17aa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"otd5JgKzB9Er3GNtflI6R19CyNQDKdwwCa+1p547k2T6P+vgnyvIzsY4HLrvWK44FWHbBSJZa2BPoqFm7LstDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T18:45:55.942797Z","bundle_sha256":"37518498d2f4578c9a149c2530dd16b1005c90877cdbd6324fb21a5d3c68d299"}}