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We also show that if a map $\\phi$ is an exposed point of $\\fP$ then either $\\phi$ is rank 1 non-increasing or $\\rank\\phi(P)>1$ for any one-dimensional projection $P\\in\\fB(\\cK)$."},"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":"1103.3497","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-03-17T19:55:07Z","cross_cats_sorted":["quant-ph"],"title_canon_sha256":"7e7bea527521312aa07d84b223e0e0c1c9e9314547e850d5ed85075fd02d3e57","abstract_canon_sha256":"890d89cb8fea6aadda063591558c52079e819048192f13f31c03b1950f59f4a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:18.761625Z","signature_b64":"nSOXztbXC86d4RD6KVii+frHTd7ByVchMRvEp0pilr1GtGOi/b69NAoyL7UgV5x/bLHhVeAQonq1Tjzxx3EqDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ebb2bad387b83e5d42d7e763099d019ba04e7d79d78c41e8a7ce9bc882ff784","last_reissued_at":"2026-05-18T02:49:18.761001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:18.761001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rank properties of exposed positive maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["quant-ph"],"primary_cat":"math.FA","authors_text":"Marcin Marciniak","submitted_at":"2011-03-17T19:55:07Z","abstract_excerpt":"Let $\\cK$ and $\\cH$ be finite dimensional Hilbert spaces and let $\\fP$ denote the cone of all positive linear maps acting from $\\fB(\\cK)$ into $\\fB(\\cH)$. 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