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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","authors_text":"Shokoufe Faraji, Zahra Baghali Khanian","cross_cats":["cs.IT","math.IT"],"headline":"A simple algebraic condition on quantum maps makes the Rényi-2 entanglement of purification additive.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2026-05-14T21:41:31Z","title":"Additivity Results for the R\\'enyi-2 Entanglement of Purification"},"references":{"count":32,"internal_anchors":4,"resolved_work":32,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"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","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"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","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"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","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Terhal, Michał Horodecki, Debbie W","work_id":"60c0c729-7501-4aa7-becc-e59f4b9057ae","year":2002},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"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","year":2002}],"snapshot_sha256":"6101204c39581174df9f7c0e9b61ded1ffc7d2a5ffd8a5022cfe57a438da2144"},"source":{"id":"2605.15439","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T14:44:04.551487Z","id":"040f44e3-3ded-4c91-9a37-34e3556e8332","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"A simple algebraic condition on quantum maps makes the Rényi-2 entanglement of purification additive.","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.","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."}},"verdict_id":"040f44e3-3ded-4c91-9a37-34e3556e8332"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:66902caec1d03fc0dada5cd1568b8542483299130c2358ca6195a8ff82d73bb8","target":"record","created_at":"2026-05-20T00:00:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7dc1be3dab84149465bdd8d114034adbb019f633da3f7c9bc3fad755b23b635c","cross_cats_sorted":["cs.IT","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2026-05-14T21:41:31Z","title_canon_sha256":"171ed99c43984e256b1a138673220b7c5d61ea7c3e100d77b7667b2da7b48bc3"},"schema_version":"1.0","source":{"id":"2605.15439","kind":"arxiv","version":1}},"canonical_sha256":"23bc520c2209a5aceeb1d1e8ebbbb4e90d91405a8e6690b6ca3813d109cce031","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"23bc520c2209a5aceeb1d1e8ebbbb4e90d91405a8e6690b6ca3813d109cce031","first_computed_at":"2026-05-20T00:00:58.626788Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:00:58.626788Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4oJ3hk2tT6shpnALLfPN4Lq0JnahMpX5flZlnMXWtxAAA3QNs8nX8+4B0y++WBlBdRCBsoDq+fJrHHohevGdAQ==","signature_status":"signed_v1","signed_at":"2026-05-20T00:00:58.627486Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.15439","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:66902caec1d03fc0dada5cd1568b8542483299130c2358ca6195a8ff82d73bb8","sha256:c0e392618219a1b93e3151a9aef026ed24463ede9acb2cd886d78d0d69597625"],"state_sha256":"de5b10cec6e1c85e7bc934657de72dfc35951f57434a4f9824b2351d49aba163"}