{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1995:NTHEFXPTLQEDSLE5LT6ELAMLED","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":"9f92a932eae6df7eda5b7f23c5930efeab74d209c3af8c82bd1f236a44cbbf20","cross_cats_sorted":[],"license":"","primary_cat":"quant-ph","submitted_at":"1995-12-30T21:31:00Z","title_canon_sha256":"fd00763dab4a7b56a06e00397974cf7411bb891f9125bf54a95cec425d0730ce"},"schema_version":"1.0","source":{"id":"quant-ph/9512032","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"quant-ph/9512032","created_at":"2026-07-04T15:59:58Z"},{"alias_kind":"arxiv_version","alias_value":"quant-ph/9512032v2","created_at":"2026-07-04T15:59:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.quant-ph/9512032","created_at":"2026-07-04T15:59:58Z"},{"alias_kind":"pith_short_12","alias_value":"NTHEFXPTLQED","created_at":"2026-07-04T15:59:58Z"},{"alias_kind":"pith_short_16","alias_value":"NTHEFXPTLQEDSLE5","created_at":"2026-07-04T15:59:58Z"},{"alias_kind":"pith_short_8","alias_value":"NTHEFXPT","created_at":"2026-07-04T15:59:58Z"}],"graph_snapshots":[{"event_id":"sha256:a73e164954c06efb14ea221762e36603e23a07c5af5dbe01f4daca3dfa777bde","target":"graph","created_at":"2026-07-04T15:59:58Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/quant-ph/9512032/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n = 1 - 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log (1-p). Upper bounds on this asymptotic rate are given.","authors_text":"A. R. Calderbank, Peter W. Shor (AT&T Research)","cross_cats":[],"headline":"","license":"","primary_cat":"quant-ph","submitted_at":"1995-12-30T21:31:00Z","title":"Good Quantum Error-Correcting Codes Exist"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"quant-ph/9512032","kind":"arxiv","version":2},"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":"sha256:4a83ff574763a2f46854c35e7e07a42592791d15aa012fb263100d670165ddd0","target":"record","created_at":"2026-07-04T15:59:58Z","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":"9f92a932eae6df7eda5b7f23c5930efeab74d209c3af8c82bd1f236a44cbbf20","cross_cats_sorted":[],"license":"","primary_cat":"quant-ph","submitted_at":"1995-12-30T21:31:00Z","title_canon_sha256":"fd00763dab4a7b56a06e00397974cf7411bb891f9125bf54a95cec425d0730ce"},"schema_version":"1.0","source":{"id":"quant-ph/9512032","kind":"arxiv","version":2}},"canonical_sha256":"6cce42ddf35c08392c9d5cfc45818b20ea6f54cb73796fd290e7febc929abf55","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6cce42ddf35c08392c9d5cfc45818b20ea6f54cb73796fd290e7febc929abf55","first_computed_at":"2026-07-04T15:59:58.199829Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:59:58.199829Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BLDTxdp96fz1AL+TBZt61pOZJSej1FjA7/Ep0UNO3T3qkN4xgVDlsVvFgA4sRTieDxGa0ND0QtOYACJG4QSmBA==","signature_status":"signed_v1","signed_at":"2026-07-04T15:59:58.200190Z","signed_message":"canonical_sha256_bytes"},"source_id":"quant-ph/9512032","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a83ff574763a2f46854c35e7e07a42592791d15aa012fb263100d670165ddd0","sha256:a73e164954c06efb14ea221762e36603e23a07c5af5dbe01f4daca3dfa777bde"],"state_sha256":"6e52d87dd0b92417b0667a2af49a26da545270d7c6aed3010ff07b0124483895"}