{"paper":{"title":"Additivity Results for the R\\'enyi-2 Entanglement of Purification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A simple algebraic condition on quantum maps makes the Rényi-2 entanglement of purification additive.","cross_cats":["cs.IT","math.IT"],"primary_cat":"quant-ph","authors_text":"Shokoufe Faraji, Zahra Baghali Khanian","submitted_at":"2026-05-14T21:41:31Z","abstract_excerpt":"We reformulate the R\\'enyi entanglement of purification as a constrained minimum output R\\'enyi entropy problem. Equivalently, for $p>1$, this formulation can be expressed in terms of a constrained maximal output Schatten $p$-norm. More precisely, for a completely positive map $\\Omega:L(B')\\to L(A)$, we consider the quantity $\\upsilon_p(\\Omega)$ defined by optimizing $\\|(\\Omega\\otimes \\mathrm{id}_E)(\\sigma^{B'E})\\|_p$ over all bipartite states $\\sigma^{B'E}$ whose $B'$-marginal is maximally mixed. We focus on the case $p=2$. First, we compute $\\upsilon_2$ for the transpose-depolarizing channel"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Whenever a completely positive map N:L(B')→L(A) satisfies N† ∘ N = a id_A + b Tr[·] I_d for constants a,b ≥ 0, the quantity υ₂(N) is multiplicative under tensor powers; this implies additivity for the associated Rényi-2 entanglement of purification.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The reformulation of the Rényi entanglement of purification as the constrained maximal output Schatten p-norm problem (for p=2) is equivalent to the original definition and preserves the relevant multiplicativity properties.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves that υ₂(Ω) is multiplicative for CP maps satisfying N† ∘ N = a id + b Tr[·]I, including depolarizing and transpose-depolarizing channels, implying additivity of Rényi-2 entanglement of purification.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A simple algebraic condition on quantum maps makes the Rényi-2 entanglement of purification additive.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1db4a8cd7e89df8675b74a86fd485a50e6073c2bd92d7903d5df27aed3684ca0"},"source":{"id":"2605.15439","kind":"arxiv","version":1},"verdict":{"id":"040f44e3-3ded-4c91-9a37-34e3556e8332","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T14:44:04.551487Z","strongest_claim":"Whenever a completely positive map N:L(B')→L(A) satisfies N† ∘ N = a id_A + b Tr[·] I_d for constants a,b ≥ 0, the quantity υ₂(N) is multiplicative under tensor powers; this implies additivity for the associated Rényi-2 entanglement of purification.","one_line_summary":"Proves that υ₂(Ω) is multiplicative for CP maps satisfying N† ∘ N = a id + b Tr[·]I, including depolarizing and transpose-depolarizing channels, implying additivity of Rényi-2 entanglement of purification.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The reformulation of the Rényi entanglement of purification as the constrained maximal output Schatten p-norm problem (for p=2) is equivalent to the original definition and preserves the relevant multiplicativity properties.","pith_extraction_headline":"A simple algebraic condition on quantum maps makes the Rényi-2 entanglement of purification additive."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15439/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"cited_work_retraction","ran_at":"2026-05-19T15:54:25.021646Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T15:50:27.837917Z","status":"completed","version":"0.1.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T15:01:38.305750Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T15:01:17.664599Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.122150Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.687834Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"28fe2dad2f1f66cc72a832060f419e24d9b849f5bb879656319ff574a576764e"},"references":{"count":32,"sample":[{"doi":"","year":null,"title":"In order to obtain sharp bounds (and later to identify candidate optimizers), it is essential to know the spectrum ofτAA′ explicitly","work_id":"5ff807ca-50bf-4991-9931-ffe6f0d5afcf","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"→ L (A)is CP,Λ : L(B1 ⊗B 2) → L (E)is CPTP, andΦ B′ 1B1 d andΦ B′ 2B2 d are normalized maximally entangled states of dimensiond. Finally, we obtain the following explicit form of(Γc t)† in Lemma 40: (","work_id":"6f72709d-3fb2-4be8-86f1-e6dec9bb161e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"→ L(A)is the adjoint of the complementary channel ofΓt. This is a CP map obtained as (Γc t)†(Y) = Tr B′ 2(St Y S t), where St = (a+ +a −)IB′ 1B′ 2 + (a+ −a −)ΠB′ 1B′ 2 .(C16) HereΠ B′ 1B′ 2 is the fli","work_id":"58666546-8280-4e74-a623-c9f9c0972e91","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"Terhal, Michał Horodecki, Debbie W","work_id":"60c0c729-7501-4aa7-becc-e59f4b9057ae","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"R. F. Werner and A. S. Holevo. Counterexample to an additivity conjecture for output purity of quantum channels.Journal of Mathematical Physics, 43(9):4353–4357, 2002","work_id":"f3424188-16ad-4e17-bb0a-bf7384be64d9","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"6101204c39581174df9f7c0e9b61ded1ffc7d2a5ffd8a5022cfe57a438da2144","internal_anchors":4},"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"}