A factorization property of positive maps on C*-algebras

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 A(i), C-i be unital C*-algebras and let alpha(i) be positive linear maps from A(i) to C-i i = 1, 2. We obtain conditions under which any positive map beta from the minimal C*-tensor product A(1) circle times(min) A(2) to C-1 circle times(min) C-2, such that alpha(1) circle times alpha(2) >= beta, factorizes as beta = gamma circle times alpha(2) for some positive map gamma. In particular, we show that when alpha(i) : -> B(H-i) are completely positive (CP) maps for some Hilbert spaces H-i (i = 1, 2), and alpha(2) is a pure CP map and beta is a CP map so that alpha(1) circle times alpha(2) - beta is also CP, then beta = gamma circle times alpha(2) for some CP map gamma. We show that a similar result holds in the context of positive linear maps when A(2) = C-2 = B(H) and alpha(2) = 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 tau from a unital C*-algebra A to a C*-algebra C, if tau circle times id(k) is decomposable for some k >= 2, where id(k) is the identity map on the algebra M-k (C) of k x k matrices, then tau is CP.