| Abstrakt: |
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 α i be positive linear maps from A i to C i , i = 1 , 2. We obtain conditions under which any positive map β from the minimal C*-tensor product A 1 ⊗ min A 2 to C 1 ⊗ min C 2 , such that α 1 ⊗ α 2 ≥ β , factorizes as β = γ ⊗ α 2 for some positive map γ. In particular, we show that when α i : A i → B (ℋ i) are completely positive (CP) maps for some Hilbert spaces ℋ i (i = 1 , 2) , and α 2 is a pure CP map and β is a CP map so that α 1 ⊗ α 2 − β is also CP, then β = γ ⊗ α 2 for some CP map γ. We show that a similar result holds in the context of positive linear maps when A 2 = C 2 = B (ℋ) and α 2 = i d. 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 ⊗ i d k is decomposable for some k ≥ 2 , where i d k is the identity map on the algebra M k (ℂ) of k × k matrices, then τ is CP. [ABSTRACT FROM AUTHOR] |
