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We first show that a rank-1 PVM on $\\mathbb{C}^d\\otimes\\mathbb{C}^d$ containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most $d$ if and only if it forms a maximally entangled basis corresponding to a nice unitary error "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"a rank-1 PVM on C^d ⊗ C^d containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most d if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The PVM must be rank-1 and contain at least one element of maximal Schmidt rank; the characterization is stated only under this restriction and may not extend without it.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Rank-1 PVMs on two qudits with a maximal-Schmidt-rank element are localizable with Schmidt number at most d exactly when they correspond to nice unitary error bases; the two-qubit case is fully classified, resolving a prior conjecture.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A rank-1 joint measurement on two d-dimensional systems can be localized with entanglement of Schmidt number at most d if and only if its elements form a maximally entangled basis from a nice unitary error basis.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2ee871af0a090430d2e1dcf0f786b1afd0da4e0fb0d7788452f5a95fc808b63b"},"source":{"id":"2601.02660","kind":"arxiv","version":2},"verdict":{"id":"ad14b600-fec9-410d-8538-6ecfd9824997","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T17:46:25.758594Z","strongest_claim":"a rank-1 PVM on C^d ⊗ C^d containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most d if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis","one_line_summary":"Rank-1 PVMs on two qudits with a maximal-Schmidt-rank element are localizable with Schmidt number at most d exactly when they correspond to nice unitary error bases; the two-qubit case is fully classified, resolving a prior conjecture.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The PVM must be rank-1 and contain at least one element of maximal Schmidt rank; the characterization is stated only under this restriction and may not extend without it.","pith_extraction_headline":"A rank-1 joint measurement on two d-dimensional systems can be localized with entanglement of Schmidt number at most d if and only if its elements form a maximally entangled basis from a nice unitary error basis."},"references":{"count":38,"sample":[{"doi":"","year":null,"title":"and partially entangled bases, such as { |00⟩, |11⟩, |01⟩ + |10⟩ √ 2 , |01⟩ − |10⟩√ 2 } (named “pBSM” in [14]). 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