A factorization property of positive maps on C∗-algebras

Article Type

Research Article

Publication Title

International Journal of Quantum Information


The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let Ai, Ci be unital C∗-algebras and let αi be positive linear maps from Ai to Ci, i=1,2. We obtain conditions under which any positive map β from the minimal C∗-tensor product A1⊗minA2 to C1⊗minC2, such that α1⊗α2≥β, factorizes as β=γ⊗α2 for some positive map γ. In particular, we show that when αi:Ai→B(Hi) are completely positive (CP) maps for some Hilbert spaces Hi (i=1,2), and α2 is a pure CP map and β is a CP map so that α1⊗α2-β is also CP, then β=γ⊗α2 for some CP map γ. We show that a similar result holds in the context of positive linear maps when A2=C2=B(H) and α2=id. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map τ from a unital C∗-algebra A to a C∗-algebra C, if τ⊗idk is decomposable for some k≥2, where idk is the identity map on the algebra Mk(ℂ) of k×k matrices, then τ is CP.



Publication Date



Open Access, Green