{"paper":{"title":"Localization of joint quantum measurements on $\\mathbb{C}^d \\otimes \\mathbb{C}^d$ by entangled resources with Schmidt number at most $d$","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","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.","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Jisho Miyazaki, Seiseki Akibue","submitted_at":"2026-01-06T02:18:17Z","abstract_excerpt":"Localizable measurements are joint quantum measurements that can be implemented using only non-adaptive local operations and shared entanglement. We provide a protocol-independent characterization of localizable projection-valued measures (PVMs) by exploiting algebraic structures that any such measurement must satisfy. 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]). Among iso-entangled bases, the higher-dimensional generalization [27] of the elegant joi","work_id":"eb23d05e-780c-4fe9-85bf-f358bcc228a3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"ideal measurement","work_id":"42439188-182d-457b-b8ba-6ea6c5730f37","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"which subspace","work_id":"90a1add5-97fb-48c1-806b-a155a2f9db58","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"N. Gisin and F. Del Santo, Towards a measurement the- ory in QFT: ”Impossible” quantum measurements are possible but not ideal, Quantum 8, 1267 (2024)","work_id":"f857735b-e645-406f-af48-aa42a7abb729","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1980,"title":"Y. Aharonov and D. Z. Albert, States and ob- servables in relativistic quantum field theories, Phys. Rev. D 21, 3316 (1980)","work_id":"9f09e228-5955-41c0-b8c1-9da9f617dd73","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":38,"snapshot_sha256":"98efd0c3701249953cee91ab8d15260c8eae044cd7e7274dbfcca7058c2abb59","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"e9c1a6a8d0c19a3649eacbf15ba0cd26c181f04628a5a93daba1df85581f1234"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}