{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:BZIGUWUMEAUUS3DYCA7PAQJR2J","short_pith_number":"pith:BZIGUWUM","canonical_record":{"source":{"id":"1803.09788","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-03-26T18:50:50Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"85ae7557c44b9e85bda8b19def5814f1feb188a9d27472a27d47b8e20b2f85d1","abstract_canon_sha256":"9637beb7fad06da018e49c9aeb97154ebcf5cc850e6318c97ff25870093d5ad3"},"schema_version":"1.0"},"canonical_sha256":"0e506a5a8c2029496c78103ef04131d263c8f810105ad327e374ee4aaf4122f6","source":{"kind":"arxiv","id":"1803.09788","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.09788","created_at":"2026-05-17T23:48:40Z"},{"alias_kind":"arxiv_version","alias_value":"1803.09788v2","created_at":"2026-05-17T23:48:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.09788","created_at":"2026-05-17T23:48:40Z"},{"alias_kind":"pith_short_12","alias_value":"BZIGUWUMEAUU","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"BZIGUWUMEAUUS3DY","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"BZIGUWUM","created_at":"2026-05-18T12:32:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:BZIGUWUMEAUUS3DYCA7PAQJR2J","target":"record","payload":{"canonical_record":{"source":{"id":"1803.09788","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-03-26T18:50:50Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"85ae7557c44b9e85bda8b19def5814f1feb188a9d27472a27d47b8e20b2f85d1","abstract_canon_sha256":"9637beb7fad06da018e49c9aeb97154ebcf5cc850e6318c97ff25870093d5ad3"},"schema_version":"1.0"},"canonical_sha256":"0e506a5a8c2029496c78103ef04131d263c8f810105ad327e374ee4aaf4122f6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:40.640658Z","signature_b64":"AZH5oRaZVGQn+87gKPmy0p6/eRkUabd6UIZ2VO1a8EKj8Lnxc7qBAYwl8fr9OBGxW9CkCyExGuUjJyzWfWrNCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e506a5a8c2029496c78103ef04131d263c8f810105ad327e374ee4aaf4122f6","last_reissued_at":"2026-05-17T23:48:40.640204Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:40.640204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.09788","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":"2026-05-17T23:48:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kMo4qgKgNx0DnED0IE1K01xni1ev+uLC+F9f498k+j5n/B2zbdQxaTPPvxcw//akXcUJc8sm8umx0Dj4DIn7Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:41:26.325852Z"},"content_sha256":"5e2445638e9e4d1841f83a94bcb68fbf874e899c641b95bd1ecd1c6b82dd7b98","schema_version":"1.0","event_id":"sha256:5e2445638e9e4d1841f83a94bcb68fbf874e899c641b95bd1ecd1c6b82dd7b98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:BZIGUWUMEAUUS3DYCA7PAQJR2J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Highly entangled tensors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.OC","authors_text":"Harm Derksen, Visu Makam","submitted_at":"2018-03-26T18:50:50Z","abstract_excerpt":"A geometric measure for the entanglement of a unit length tensor $T \\in (\\mathbb{C}^n)^{\\otimes k}$ is given by $- 2 \\log_2 ||T||_\\sigma$, where $||.||_\\sigma$ denotes the spectral norm. A simple induction gives an upper bound of $(k-1) \\log_2(n)$ for the entanglement. We show the existence of tensors with entanglement larger than $k \\log_2(n) - \\log_2(k) - o(\\log_2(k))$. Friedland and Kemp have similar results in the case of symmetric tensors. Our techniques give improvements in this case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09788","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":"2026-05-17T23:48:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A9GfDAfKk7/RWGOzDUF3FMKLYgidtuUYWIcRCRtTTaApLKDjExuRi5KLOa6FS/a1sw7R1AbXrIxq4UsxhOzXAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:41:26.326196Z"},"content_sha256":"fe302d8dec4f54b8e90204bfba005fcc2e997aaccf6b63347d6a89c593f4acc4","schema_version":"1.0","event_id":"sha256:fe302d8dec4f54b8e90204bfba005fcc2e997aaccf6b63347d6a89c593f4acc4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BZIGUWUMEAUUS3DYCA7PAQJR2J/bundle.json","state_url":"https://pith.science/pith/BZIGUWUMEAUUS3DYCA7PAQJR2J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BZIGUWUMEAUUS3DYCA7PAQJR2J/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":"2026-06-04T19:41:26Z","links":{"resolver":"https://pith.science/pith/BZIGUWUMEAUUS3DYCA7PAQJR2J","bundle":"https://pith.science/pith/BZIGUWUMEAUUS3DYCA7PAQJR2J/bundle.json","state":"https://pith.science/pith/BZIGUWUMEAUUS3DYCA7PAQJR2J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BZIGUWUMEAUUS3DYCA7PAQJR2J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:BZIGUWUMEAUUS3DYCA7PAQJR2J","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":"9637beb7fad06da018e49c9aeb97154ebcf5cc850e6318c97ff25870093d5ad3","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-03-26T18:50:50Z","title_canon_sha256":"85ae7557c44b9e85bda8b19def5814f1feb188a9d27472a27d47b8e20b2f85d1"},"schema_version":"1.0","source":{"id":"1803.09788","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.09788","created_at":"2026-05-17T23:48:40Z"},{"alias_kind":"arxiv_version","alias_value":"1803.09788v2","created_at":"2026-05-17T23:48:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.09788","created_at":"2026-05-17T23:48:40Z"},{"alias_kind":"pith_short_12","alias_value":"BZIGUWUMEAUU","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"BZIGUWUMEAUUS3DY","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"BZIGUWUM","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:fe302d8dec4f54b8e90204bfba005fcc2e997aaccf6b63347d6a89c593f4acc4","target":"graph","created_at":"2026-05-17T23:48:40Z","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":"A geometric measure for the entanglement of a unit length tensor $T \\in (\\mathbb{C}^n)^{\\otimes k}$ is given by $- 2 \\log_2 ||T||_\\sigma$, where $||.||_\\sigma$ denotes the spectral norm. A simple induction gives an upper bound of $(k-1) \\log_2(n)$ for the entanglement. We show the existence of tensors with entanglement larger than $k \\log_2(n) - \\log_2(k) - o(\\log_2(k))$. Friedland and Kemp have similar results in the case of symmetric tensors. Our techniques give improvements in this case.","authors_text":"Harm Derksen, Visu Makam","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-03-26T18:50:50Z","title":"Highly entangled tensors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09788","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":"sha256:5e2445638e9e4d1841f83a94bcb68fbf874e899c641b95bd1ecd1c6b82dd7b98","target":"record","created_at":"2026-05-17T23:48:40Z","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":"9637beb7fad06da018e49c9aeb97154ebcf5cc850e6318c97ff25870093d5ad3","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-03-26T18:50:50Z","title_canon_sha256":"85ae7557c44b9e85bda8b19def5814f1feb188a9d27472a27d47b8e20b2f85d1"},"schema_version":"1.0","source":{"id":"1803.09788","kind":"arxiv","version":2}},"canonical_sha256":"0e506a5a8c2029496c78103ef04131d263c8f810105ad327e374ee4aaf4122f6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e506a5a8c2029496c78103ef04131d263c8f810105ad327e374ee4aaf4122f6","first_computed_at":"2026-05-17T23:48:40.640204Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:40.640204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AZH5oRaZVGQn+87gKPmy0p6/eRkUabd6UIZ2VO1a8EKj8Lnxc7qBAYwl8fr9OBGxW9CkCyExGuUjJyzWfWrNCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:40.640658Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.09788","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e2445638e9e4d1841f83a94bcb68fbf874e899c641b95bd1ecd1c6b82dd7b98","sha256:fe302d8dec4f54b8e90204bfba005fcc2e997aaccf6b63347d6a89c593f4acc4"],"state_sha256":"79e947f9dd82fc77ef81cd84b399ab35dedbe362c9d5813b40b81dc8a2661111"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NkfK/sJRoHFI1QRScd51+2tEKXLh3SVEV0KLHGozbzooAzDsJdAkJhTOnV3dJYA8Tx0R0lf8UAGneWr/cyHzAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T19:41:26.328099Z","bundle_sha256":"e85da1cd7c9a8d56ab922d018ea1ca1dbd4362c65f14f3e99cda6cd96c44fbc0"}}