{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:UL5DPQDORP6CRPBBWOBT5IMQFG","short_pith_number":"pith:UL5DPQDO","canonical_record":{"source":{"id":"1309.4177","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2013-09-17T04:15:57Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a7bdd72204d0bc5126e4de7e9dd6162b403afd95af0c393869b8a53e5f050d39","abstract_canon_sha256":"2e2a4b1d26e1845d3945ea3379d0e09a3157fde3bdbc4a20a56b2baf1c1ce9a9"},"schema_version":"1.0"},"canonical_sha256":"a2fa37c06e8bfc28bc21b3833ea19029a7e9f748241267b8b2d27242573dce62","source":{"kind":"arxiv","id":"1309.4177","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4177","created_at":"2026-05-18T03:12:32Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4177v2","created_at":"2026-05-18T03:12:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4177","created_at":"2026-05-18T03:12:32Z"},{"alias_kind":"pith_short_12","alias_value":"UL5DPQDORP6C","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"UL5DPQDORP6CRPBB","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"UL5DPQDO","created_at":"2026-05-18T12:28:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:UL5DPQDORP6CRPBBWOBT5IMQFG","target":"record","payload":{"canonical_record":{"source":{"id":"1309.4177","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2013-09-17T04:15:57Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a7bdd72204d0bc5126e4de7e9dd6162b403afd95af0c393869b8a53e5f050d39","abstract_canon_sha256":"2e2a4b1d26e1845d3945ea3379d0e09a3157fde3bdbc4a20a56b2baf1c1ce9a9"},"schema_version":"1.0"},"canonical_sha256":"a2fa37c06e8bfc28bc21b3833ea19029a7e9f748241267b8b2d27242573dce62","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:32.278789Z","signature_b64":"ofm8dOF/t8m+jQeXSh5+GyAYjsdhwkX3aAK3uWtVqfao2cELocW81ilmQtDPzuEOtVQ0fDLTiF/JetLlr032Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2fa37c06e8bfc28bc21b3833ea19029a7e9f748241267b8b2d27242573dce62","last_reissued_at":"2026-05-18T03:12:32.278106Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:32.278106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.4177","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q8lDa2hiwjCxv9lSjLgL1sSztjpHJ6iDhyZ/B0RTOHo+jaOR2cnSMxPWZoUzJ5Y6sVO4CUCCKdTe9FtqHrHSDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T17:36:09.972833Z"},"content_sha256":"8af28e4742109347ce8c5df2f8f4b3883068c4451914c0278d5ed07496d95d8a","schema_version":"1.0","event_id":"sha256:8af28e4742109347ce8c5df2f8f4b3883068c4451914c0278d5ed07496d95d8a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:UL5DPQDORP6CRPBBWOBT5IMQFG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The number of product vectors and their partial conjugates in a pair of spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"quant-ph","authors_text":"Joohan Na","submitted_at":"2013-09-17T04:15:57Z","abstract_excerpt":"Let $D$ and $E$ be subspaces of the tensor product of the finite-dimensional Hilbert spaces $\\mathbb{C}^m \\otimes \\mathbb{C}^n$. We show that the number of product vectors in $D$ with their partial conjugates in $E$ is uniformly bounded depending only on $m$ and $n$ whenever it is finite. We also give an upper bound in qubit-qunit case which we expect to be sharp."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4177","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UisNh8zHbedZOI4XTa1CiqBezaO7GMZWBMwk2DG55uPd6hXl5oEMnd6I5syJpZyhELyDAw2kmt8zA/eO1FCWAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T17:36:09.973464Z"},"content_sha256":"dcd92853416e55848a471ce4d65eb6fe9b873c741093fe9d0b05748002771bc7","schema_version":"1.0","event_id":"sha256:dcd92853416e55848a471ce4d65eb6fe9b873c741093fe9d0b05748002771bc7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UL5DPQDORP6CRPBBWOBT5IMQFG/bundle.json","state_url":"https://pith.science/pith/UL5DPQDORP6CRPBBWOBT5IMQFG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UL5DPQDORP6CRPBBWOBT5IMQFG/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":"2026-05-27T17:36:09Z","links":{"resolver":"https://pith.science/pith/UL5DPQDORP6CRPBBWOBT5IMQFG","bundle":"https://pith.science/pith/UL5DPQDORP6CRPBBWOBT5IMQFG/bundle.json","state":"https://pith.science/pith/UL5DPQDORP6CRPBBWOBT5IMQFG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UL5DPQDORP6CRPBBWOBT5IMQFG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:UL5DPQDORP6CRPBBWOBT5IMQFG","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":"2e2a4b1d26e1845d3945ea3379d0e09a3157fde3bdbc4a20a56b2baf1c1ce9a9","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2013-09-17T04:15:57Z","title_canon_sha256":"a7bdd72204d0bc5126e4de7e9dd6162b403afd95af0c393869b8a53e5f050d39"},"schema_version":"1.0","source":{"id":"1309.4177","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4177","created_at":"2026-05-18T03:12:32Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4177v2","created_at":"2026-05-18T03:12:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4177","created_at":"2026-05-18T03:12:32Z"},{"alias_kind":"pith_short_12","alias_value":"UL5DPQDORP6C","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"UL5DPQDORP6CRPBB","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"UL5DPQDO","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:dcd92853416e55848a471ce4d65eb6fe9b873c741093fe9d0b05748002771bc7","target":"graph","created_at":"2026-05-18T03:12:32Z","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":"Let $D$ and $E$ be subspaces of the tensor product of the finite-dimensional Hilbert spaces $\\mathbb{C}^m \\otimes \\mathbb{C}^n$. We show that the number of product vectors in $D$ with their partial conjugates in $E$ is uniformly bounded depending only on $m$ and $n$ whenever it is finite. We also give an upper bound in qubit-qunit case which we expect to be sharp.","authors_text":"Joohan Na","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2013-09-17T04:15:57Z","title":"The number of product vectors and their partial conjugates in a pair of spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4177","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":"sha256:8af28e4742109347ce8c5df2f8f4b3883068c4451914c0278d5ed07496d95d8a","target":"record","created_at":"2026-05-18T03:12:32Z","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":"2e2a4b1d26e1845d3945ea3379d0e09a3157fde3bdbc4a20a56b2baf1c1ce9a9","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2013-09-17T04:15:57Z","title_canon_sha256":"a7bdd72204d0bc5126e4de7e9dd6162b403afd95af0c393869b8a53e5f050d39"},"schema_version":"1.0","source":{"id":"1309.4177","kind":"arxiv","version":2}},"canonical_sha256":"a2fa37c06e8bfc28bc21b3833ea19029a7e9f748241267b8b2d27242573dce62","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a2fa37c06e8bfc28bc21b3833ea19029a7e9f748241267b8b2d27242573dce62","first_computed_at":"2026-05-18T03:12:32.278106Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:32.278106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ofm8dOF/t8m+jQeXSh5+GyAYjsdhwkX3aAK3uWtVqfao2cELocW81ilmQtDPzuEOtVQ0fDLTiF/JetLlr032Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:32.278789Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.4177","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8af28e4742109347ce8c5df2f8f4b3883068c4451914c0278d5ed07496d95d8a","sha256:dcd92853416e55848a471ce4d65eb6fe9b873c741093fe9d0b05748002771bc7"],"state_sha256":"bd50875341769d503ab7b13301f246006fbfdb5cffa6bed147b8fd299c826792"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ACnXGtxdOxhCt9R9goUSAm2+DQ1e02xQk5usC94XHqqV3X6f5ce0lBzbBl9EEdcuWPdPqOMrx1H0hdEhUiKbBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T17:36:09.976767Z","bundle_sha256":"ba45b99a095e3b6cbcd372a2ea27f1740d99f487d04b9d46469b0bc77a85ffe2"}}