A factorization property of positive maps on C*-algebras

Abstract

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 [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. 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 [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.