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Let ${\\mathcal M}$ be a finite dimensional von Neumann algebra and ${\\mathcal N}$ a von Neumann subalgebra if it. Let ${\\mathcal E}_\\tau$ be the tracial conditional expectation from ${\\mathcal M}$ onto ${\\mathcal N}$. For density matrices $\\rho$ and $\\sigma$ in ${\\mathcal N}$, let $\\rho_{\\mathcal N} := {\\mathcal E}_\\tau \\rho$ and $\\sigma_{\\mathcal N} := {\\mathcal E}_\\tau \\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1710.02409","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-10-06T13:58:56Z","cross_cats_sorted":["quant-ph"],"title_canon_sha256":"cb5430833ac2478ee2fa247aa1be3008bf23a2be3eb95e00bb6861502f5f13b2","abstract_canon_sha256":"4216fc06da6f1e4546f5c691ca66358d15d349689fdf56fc949eebbe1ff24964"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:43.501185Z","signature_b64":"NnM/3kFz9Joi8k0VrxXJjA/i1t4/THsvHtv7UleZ/NlVfKG2T4XyRE68rFqZkdw4ATnHHzJK8+IkWHy9GKvXCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c66267d2d3f9b68db7da5a4549c71e240a758c4e3383a034f583217a35acb080","last_reissued_at":"2026-05-17T23:44:43.500762Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:43.500762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Recovery map stability for the Data Processing Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["quant-ph"],"primary_cat":"math.OA","authors_text":"Anna Vershynina, Eric A. 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For density matrices $\\rho$ and $\\sigma$ in ${\\mathcal N}$, let $\\rho_{\\mathcal N} := {\\mathcal E}_\\tau \\rho$ and $\\sigma_{\\mathcal N} := {\\mathcal E}_\\tau \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.02409","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1710.02409","created_at":"2026-05-17T23:44:43.500817+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.02409v3","created_at":"2026-05-17T23:44:43.500817+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.02409","created_at":"2026-05-17T23:44:43.500817+00:00"},{"alias_kind":"pith_short_12","alias_value":"YZRGPUWT7G3I","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_16","alias_value":"YZRGPUWT7G3I3N62","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_8","alias_value":"YZRGPUWT","created_at":"2026-05-18T12:31:59.375834+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2402.18500","citing_title":"Conditional Independence of 1D Gibbs States with Applications to Efficient Learning","ref_index":24,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ","json":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ.json","graph_json":"https://pith.science/api/pith-number/YZRGPUWT7G3I3N62LJCUTRY6EQ/graph.json","events_json":"https://pith.science/api/pith-number/YZRGPUWT7G3I3N62LJCUTRY6EQ/events.json","paper":"https://pith.science/paper/YZRGPUWT"},"agent_actions":{"view_html":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ","download_json":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ.json","view_paper":"https://pith.science/paper/YZRGPUWT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.02409&json=true","fetch_graph":"https://pith.science/api/pith-number/YZRGPUWT7G3I3N62LJCUTRY6EQ/graph.json","fetch_events":"https://pith.science/api/pith-number/YZRGPUWT7G3I3N62LJCUTRY6EQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ/action/storage_attestation","attest_author":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ/action/author_attestation","sign_citation":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ/action/citation_signature","submit_replication":"https://pith.science/pith/YZRGPUWT7G3I3N62LJCUTRY6EQ/action/replication_record"}},"created_at":"2026-05-17T23:44:43.500817+00:00","updated_at":"2026-05-17T23:44:43.500817+00:00"}