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Namely, for a predicate $D$ on $\\{0,1,...,n\\}$ let $\\ell_0(D)\\df \\max\\{\\ell : 1\\leq\\ell\\leq n/2\\land D(\\ell)\\not\\equiv D(\\ell-1)\\}$ and $\\ell_1(D)\\df \\max\\{n-\\ell : n/2\\leq\\ell < n\\land D(\\ell)\\not\\equiv D(\\ell+1)\\}$. Then the bounded-error quantum communication complexity of $f_D(x,y) = D(|x\\cap y|)$ is equal (again, up to a logarithmic factor) to $\\sqrt{n\\ell_0(D)}+\\ell_1(D)$. In particular, the com"},"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":"quant-ph/0204025","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"quant-ph","submitted_at":"2002-04-04T22:49:34Z","cross_cats_sorted":[],"title_canon_sha256":"05f4faa098b0383735c40310a3a799c021dc5408494494a0b8e3d30bff6facb8","abstract_canon_sha256":"5d83e6e52058be63cbc16f9f2122186855ce16969e95913faa4980a15a66703c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:01.994283Z","signature_b64":"/nuRrx8WHXP1BKaAqxmCSzxP7CdGyu17R6eEpbXguwV5qBtZdCKtfZaDJ8upKEhDHfZbRMpVFWv4EHc3LsxrAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84571b2790d1f6409ea8993a99152c30a26a77e62e728be138fb439cb7780d2d","last_reissued_at":"2026-05-18T01:38:01.993758Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:01.993758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quantum communication complexity of symmetric predicates","license":"","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Alexander Razborov","submitted_at":"2002-04-04T22:49:34Z","abstract_excerpt":"We completely (that is, up to a logarithmic factor) characterize the bounded-error quantum communication complexity of every predicate $f(x,y)$ depending only on $|x\\cap y|$ ($x,y\\subseteq [n]$). Namely, for a predicate $D$ on $\\{0,1,...,n\\}$ let $\\ell_0(D)\\df \\max\\{\\ell : 1\\leq\\ell\\leq n/2\\land D(\\ell)\\not\\equiv D(\\ell-1)\\}$ and $\\ell_1(D)\\df \\max\\{n-\\ell : n/2\\leq\\ell < n\\land D(\\ell)\\not\\equiv D(\\ell+1)\\}$. Then the bounded-error quantum communication complexity of $f_D(x,y) = D(|x\\cap y|)$ is equal (again, up to a logarithmic factor) to $\\sqrt{n\\ell_0(D)}+\\ell_1(D)$. 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