{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:1996:QW6LWVF2HEGZSUABEVEDCAUHLU","short_pith_number":"pith:QW6LWVF2","schema_version":"1.0","canonical_sha256":"85bcbb54ba390d99500125483102875d10badbda05ac57f0e8e133db4c5d9361","source":{"kind":"arxiv","id":"quant-ph/9610040","version":1},"attestation_state":"computed","paper":{"title":"Quantum MacWilliams Identities","license":"","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Peter Shor, Raymond Laflamme","submitted_at":"1996-10-25T21:32:14Z","abstract_excerpt":"We derive a relationship between two different notions of fidelity (entanglement fidelity and average fidelity) for a completely depolarizing quantum channel. This relationship gives rise to a quantum analog of the MacWilliams identities in classical coding theory. These identities relate the weight enumerator of a code to the one of its dual and, with linear programming techniques, provided a powerful tool to investigate the possible existence of codes. The same techniques can be adapted to the quantum case. We give examples of their power."},"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":"quant-ph/9610040","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"quant-ph","submitted_at":"1996-10-25T21:32:14Z","cross_cats_sorted":[],"title_canon_sha256":"2de63f420bc2f64927df6bfd13d9a980ae576ee108dfe7508a5d731dc8ad383c","abstract_canon_sha256":"12df8876a2098effbd4c21abcf4625490e65dac9877e0817f2f67f3ae58d2571"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:47:05.636168Z","signature_b64":"vFAh1bTfwza0lOHKZ1OfB3/YM6TmTouKBOR0Cp/RLQw9UJ4I8gIvohlmWoJNwkxtyncZi5qcckpa6xf8r22WBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85bcbb54ba390d99500125483102875d10badbda05ac57f0e8e133db4c5d9361","last_reissued_at":"2026-07-04T14:47:05.635722Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:47:05.635722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quantum MacWilliams Identities","license":"","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Peter Shor, Raymond Laflamme","submitted_at":"1996-10-25T21:32:14Z","abstract_excerpt":"We derive a relationship between two different notions of fidelity (entanglement fidelity and average fidelity) for a completely depolarizing quantum channel. This relationship gives rise to a quantum analog of the MacWilliams identities in classical coding theory. These identities relate the weight enumerator of a code to the one of its dual and, with linear programming techniques, provided a powerful tool to investigate the possible existence of codes. The same techniques can be adapted to the quantum case. We give examples of their power."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"quant-ph/9610040","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/quant-ph/9610040/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"quant-ph/9610040","created_at":"2026-07-04T14:47:05.635786+00:00"},{"alias_kind":"arxiv_version","alias_value":"quant-ph/9610040v1","created_at":"2026-07-04T14:47:05.635786+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.quant-ph/9610040","created_at":"2026-07-04T14:47:05.635786+00:00"},{"alias_kind":"pith_short_12","alias_value":"QW6LWVF2HEGZ","created_at":"2026-07-04T14:47:05.635786+00:00"},{"alias_kind":"pith_short_16","alias_value":"QW6LWVF2HEGZSUAB","created_at":"2026-07-04T14:47:05.635786+00:00"},{"alias_kind":"pith_short_8","alias_value":"QW6LWVF2","created_at":"2026-07-04T14:47:05.635786+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2606.04940","citing_title":"Convergence rates of Sum-of-Hermitian-Squares Hierarchies for the Pauli algebra","ref_index":67,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU","json":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU.json","graph_json":"https://pith.science/api/pith-number/QW6LWVF2HEGZSUABEVEDCAUHLU/graph.json","events_json":"https://pith.science/api/pith-number/QW6LWVF2HEGZSUABEVEDCAUHLU/events.json","paper":"https://pith.science/paper/QW6LWVF2"},"agent_actions":{"view_html":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU","download_json":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU.json","view_paper":"https://pith.science/paper/QW6LWVF2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=quant-ph/9610040&json=true","fetch_graph":"https://pith.science/api/pith-number/QW6LWVF2HEGZSUABEVEDCAUHLU/graph.json","fetch_events":"https://pith.science/api/pith-number/QW6LWVF2HEGZSUABEVEDCAUHLU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU/action/storage_attestation","attest_author":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU/action/author_attestation","sign_citation":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU/action/citation_signature","submit_replication":"https://pith.science/pith/QW6LWVF2HEGZSUABEVEDCAUHLU/action/replication_record"}},"created_at":"2026-07-04T14:47:05.635786+00:00","updated_at":"2026-07-04T14:47:05.635786+00:00"}