bibkey: aubrun2024completely authors: Guillaume Aubrun, Kenneth R. Davidson, Alexander Müller-Hermes, Vern I. Paulsen and Mizanur Rahaman year: 2024 title: “Completely bounded norms of k-positive maps” doi: 10.1112/jlms.12936 url: https://doi.org/10.1112/jlms.12936 claim: “Theorem 3.7: d₁(M_n) = n, i.e. every unital positive map on M_n has completely bounded norm at most n; the proof uses the separability of the block operator with identity diagonal blocks and off-diagonal block x/n for every contraction x in M_m(M_n).” strata_touched:
- D5/S3/Quantum/Entanglement/AbsoluteSeparability/ContractionBlocks license: citation-only triage: anchor
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Source: https://doi.org/10.1112/jlms.12936
DOI: 10.1112/jlms.12936 (J. London Math. Soc. 109, e12936). Text read from https://arxiv.org/abs/2401.12352v2, printed p. 9.
Theorem 3.7 reads:
For every n ⩾ 2, we have d₁(M_n) = n.
Its proof reads in part:
Fix an integer m ⩾ 1 and an element x ∈ M_m(M_n) such that ‖x‖ ⩽ 1. By [1, Corollary 8.4], the operator is separable, i.e., is a positive combination of elements of the form y ⊗ z for y ∈ M⁺_{2m} and z ∈ M⁺_n.
Reference [1] is T. Ando, Cones and norms in the tensor product of matrix spaces, Linear Algebra Appl. 379 (2004), 3–41 (not read). By the duality between separable and block-positive matrices, Theorem 3.7 is equivalent to the bound |Tr(WH)| ≤ m Tr W for block-positive W on M_m ⊗ M_n and Hermitian contractions H, i.e. to the separability of m I + H.