{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:B3QQCUJR5677677JJB4W6KKHOC","short_pith_number":"pith:B3QQCUJR","canonical_record":{"source":{"id":"1203.1364","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-03-07T01:47:24Z","cross_cats_sorted":["math.MP","quant-ph"],"title_canon_sha256":"efc05a9ae48fd7d681192342745021a48309b3a173cc5e7ea77fa1a68d10687e","abstract_canon_sha256":"8d17140f474c2502ed4ae8b22935f28e6c6708c8a8a11973ad5f7ee49e42617f"},"schema_version":"1.0"},"canonical_sha256":"0ee1015131efbfff7fe948796f294770922d5c46e025733cf8e4056a58d90df4","source":{"kind":"arxiv","id":"1203.1364","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1364","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1364v4","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1364","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"pith_short_12","alias_value":"B3QQCUJR5677","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"B3QQCUJR5677677J","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"B3QQCUJR","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:B3QQCUJR5677677JJB4W6KKHOC","target":"record","payload":{"canonical_record":{"source":{"id":"1203.1364","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-03-07T01:47:24Z","cross_cats_sorted":["math.MP","quant-ph"],"title_canon_sha256":"efc05a9ae48fd7d681192342745021a48309b3a173cc5e7ea77fa1a68d10687e","abstract_canon_sha256":"8d17140f474c2502ed4ae8b22935f28e6c6708c8a8a11973ad5f7ee49e42617f"},"schema_version":"1.0"},"canonical_sha256":"0ee1015131efbfff7fe948796f294770922d5c46e025733cf8e4056a58d90df4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:02.286954Z","signature_b64":"NhBd0iQgreXdpwmNLSNJatA8YNAWD++cnH3XnSzuUL0/nW84xGiny0lRvNZt1At5ohKsJOUMX7c3iwyZPfeRCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ee1015131efbfff7fe948796f294770922d5c46e025733cf8e4056a58d90df4","last_reissued_at":"2026-05-18T03:07:02.286510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:02.286510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.1364","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tsM5TjC7AZXswIF9eViqwJYVa3vGfcb0iZXfos2S2eThcncFjCxR2nhOqb+6+vjqbL6tMsHgL++pYhYoA01gDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:04:27.396971Z"},"content_sha256":"3fcc3921001c182812077bd7b42f47eccdfa12d272169822345ce7128cfc3e34","schema_version":"1.0","event_id":"sha256:3fcc3921001c182812077bd7b42f47eccdfa12d272169822345ce7128cfc3e34"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:B3QQCUJR5677677JJB4W6KKHOC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Properties and construction of extreme bipartite states having positive partial transpose","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","quant-ph"],"primary_cat":"math-ph","authors_text":"Dragomir Z. Djokovic, Lin Chen","submitted_at":"2012-03-07T01:47:24Z","abstract_excerpt":"We consider a bipartite quantum system H_A x H_B with M=dim H_A and N=dim H_B. We study the set E of extreme points of the compact convex set of all states having positive partial transpose (PPT) and its subsets E_r={rho in E: rank rho=r}. Our main results pertain to the subsets E_r^{M,N} of E_r consisting of states whose reduced density operators have ranks M and N, respectively. The set E_1 is just the set of pure product states. It is known that E_r^{M,N} is empty for 1< r <= min(M,N) and for r=MN. We prove that also E_{MN-1}^{M,N} is empty. Leinaas, Myrheim and Sollid have conjectured that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1364","kind":"arxiv","version":4},"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":"2026-05-18T03:07:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wn6gnImTlp/yxku2SiKmbFLphDQC5jFzBxq4OGpQ532KNovlmjJxWkJd+OALpeooq0cGTg5DSVDUjRg56M+7Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:04:27.397687Z"},"content_sha256":"7b8ffd0a635f35f546632abb688eabb518efffdae7bb057003b17be11a1ae08f","schema_version":"1.0","event_id":"sha256:7b8ffd0a635f35f546632abb688eabb518efffdae7bb057003b17be11a1ae08f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B3QQCUJR5677677JJB4W6KKHOC/bundle.json","state_url":"https://pith.science/pith/B3QQCUJR5677677JJB4W6KKHOC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B3QQCUJR5677677JJB4W6KKHOC/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":"2026-06-01T02:04:27Z","links":{"resolver":"https://pith.science/pith/B3QQCUJR5677677JJB4W6KKHOC","bundle":"https://pith.science/pith/B3QQCUJR5677677JJB4W6KKHOC/bundle.json","state":"https://pith.science/pith/B3QQCUJR5677677JJB4W6KKHOC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B3QQCUJR5677677JJB4W6KKHOC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:B3QQCUJR5677677JJB4W6KKHOC","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":"8d17140f474c2502ed4ae8b22935f28e6c6708c8a8a11973ad5f7ee49e42617f","cross_cats_sorted":["math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-03-07T01:47:24Z","title_canon_sha256":"efc05a9ae48fd7d681192342745021a48309b3a173cc5e7ea77fa1a68d10687e"},"schema_version":"1.0","source":{"id":"1203.1364","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.1364","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"arxiv_version","alias_value":"1203.1364v4","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1364","created_at":"2026-05-18T03:07:02Z"},{"alias_kind":"pith_short_12","alias_value":"B3QQCUJR5677","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"B3QQCUJR5677677J","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"B3QQCUJR","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:7b8ffd0a635f35f546632abb688eabb518efffdae7bb057003b17be11a1ae08f","target":"graph","created_at":"2026-05-18T03:07:02Z","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 consider a bipartite quantum system H_A x H_B with M=dim H_A and N=dim H_B. We study the set E of extreme points of the compact convex set of all states having positive partial transpose (PPT) and its subsets E_r={rho in E: rank rho=r}. Our main results pertain to the subsets E_r^{M,N} of E_r consisting of states whose reduced density operators have ranks M and N, respectively. The set E_1 is just the set of pure product states. It is known that E_r^{M,N} is empty for 1< r <= min(M,N) and for r=MN. We prove that also E_{MN-1}^{M,N} is empty. Leinaas, Myrheim and Sollid have conjectured that","authors_text":"Dragomir Z. Djokovic, Lin Chen","cross_cats":["math.MP","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-03-07T01:47:24Z","title":"Properties and construction of extreme bipartite states having positive partial transpose"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1364","kind":"arxiv","version":4},"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":"sha256:3fcc3921001c182812077bd7b42f47eccdfa12d272169822345ce7128cfc3e34","target":"record","created_at":"2026-05-18T03:07:02Z","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":"8d17140f474c2502ed4ae8b22935f28e6c6708c8a8a11973ad5f7ee49e42617f","cross_cats_sorted":["math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-03-07T01:47:24Z","title_canon_sha256":"efc05a9ae48fd7d681192342745021a48309b3a173cc5e7ea77fa1a68d10687e"},"schema_version":"1.0","source":{"id":"1203.1364","kind":"arxiv","version":4}},"canonical_sha256":"0ee1015131efbfff7fe948796f294770922d5c46e025733cf8e4056a58d90df4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0ee1015131efbfff7fe948796f294770922d5c46e025733cf8e4056a58d90df4","first_computed_at":"2026-05-18T03:07:02.286510Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:02.286510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NhBd0iQgreXdpwmNLSNJatA8YNAWD++cnH3XnSzuUL0/nW84xGiny0lRvNZt1At5ohKsJOUMX7c3iwyZPfeRCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:02.286954Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.1364","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fcc3921001c182812077bd7b42f47eccdfa12d272169822345ce7128cfc3e34","sha256:7b8ffd0a635f35f546632abb688eabb518efffdae7bb057003b17be11a1ae08f"],"state_sha256":"48ef8a0375bfcfc45fce67eabb259184fba0a99902d6610fb23e611e33283152"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RPhVN2gM2C0DFhLgy8GsfXH66Tt+CA5YwJRJ2GGZ5Xry0hgP9W4kIxZL245twnu1B60ZP9S7ttJYWpF3vQsoCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T02:04:27.401701Z","bundle_sha256":"96011da84994c5876ba2e47aefa7b9c298dfd13f5cd3a00f98d3cc75550942d0"}}