{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:5XQJZXBX2K5E5IAEZDXS6RP3DN","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":"63b7d06e110ca47e23d56514bd354cb14c749fefd9edfa8e592ce17472a39e52","cross_cats_sorted":["math.AG","math.DG","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-06-05T19:38:23Z","title_canon_sha256":"04d54ea91993ac952532fc65f37ab30c2d2ebf22269e3bc149aa155c82df23a9"},"schema_version":"1.0","source":{"id":"1306.1214","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.1214","created_at":"2026-05-18T01:37:21Z"},{"alias_kind":"arxiv_version","alias_value":"1306.1214v2","created_at":"2026-05-18T01:37:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.1214","created_at":"2026-05-18T01:37:21Z"},{"alias_kind":"pith_short_12","alias_value":"5XQJZXBX2K5E","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"5XQJZXBX2K5E5IAE","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"5XQJZXBX","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:0631ac806f73fbfa2111b68669e1a0b5cf16cb5f0c606b6c6e588a9d8179db52","target":"graph","created_at":"2026-05-18T01:37:21Z","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 show that for any positive integer $n$, the maps $x \\in \\mathbb{C}^n \\mapsto \\{\\left|\\langle x, z_i \\rangle \\right|^2\\}_{i=1}^{4n} \\in \\mathbb{R}^{4n}$, where $z_i$ are the columns of four $n\\times n$ unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of embeddings of $\\mathbb{C}P^{n-1}$ into $\\mathbb{R}^{4n-4}$. In particular, this implies that distribution measurements about a pure state with four generic full-rank observables are informationally complete, which is sharp for $n \\geq 6$. To complement this information-theoretic stu","authors_text":"Damien Mondragon, Vladislav Voroninski","cross_cats":["math.AG","math.DG","math.MP","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-06-05T19:38:23Z","title":"Determination of all pure quantum states from a minimal number of observables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1214","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:a97380bdbfa1e139ca4b0848e87221a96a090e73ede6f950a79a2aa59c25b713","target":"record","created_at":"2026-05-18T01:37:21Z","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":"63b7d06e110ca47e23d56514bd354cb14c749fefd9edfa8e592ce17472a39e52","cross_cats_sorted":["math.AG","math.DG","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-06-05T19:38:23Z","title_canon_sha256":"04d54ea91993ac952532fc65f37ab30c2d2ebf22269e3bc149aa155c82df23a9"},"schema_version":"1.0","source":{"id":"1306.1214","kind":"arxiv","version":2}},"canonical_sha256":"ede09cdc37d2ba4ea004c8ef2f45fb1b554b304f217c94ce58e9fed221c753a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ede09cdc37d2ba4ea004c8ef2f45fb1b554b304f217c94ce58e9fed221c753a6","first_computed_at":"2026-05-18T01:37:21.807486Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:21.807486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T1ul5XFMZxHPZkpUcCUAj4EzHlEArgNKQbTyoa1CRTqh2SPTArdSFiufAw4y2Nm7WjwzCDGERHhtNzJSKVtEDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:21.808199Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.1214","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a97380bdbfa1e139ca4b0848e87221a96a090e73ede6f950a79a2aa59c25b713","sha256:0631ac806f73fbfa2111b68669e1a0b5cf16cb5f0c606b6c6e588a9d8179db52"],"state_sha256":"488d0486c274d2e2a49c8536888294c5d303ca38d5bfa52ec47f4f392b44e16a"}