Entanglement-breaking channels with general outcome operator algebras

A unit-preserving and completely positive linear map, or a channel, Lambda : A -> A(in) between C*-algebras A and A(in) is called entanglement-breaking (EB) if omega o (Lambda circle times id(B)) is a separable state for any C*-algebras B and any state omega on the injective C*-tensor product A(in) circle times B. In this paper, we establish the equivalence of the following conditions for a channel Lambda with a quantum input space and with a general outcome C*-algebras, generalizing the known results in finite dimensions: (i) Lambda is EB; (ii) Lambda has a measurement-prepare form (Holevo form); (iii) n copies of Lambda are compatible for all 2 <= n < infinity; (iv) countably infinite copies of Lambda are compatible. By using this equivalence, we also show that the set of randomization-equivalence classes of normal EB channels with a fixed input von Neumann algebra is upper and lower Dedekind-closed, i. i.e., the supremum or infimum of any randomization-increasing or decreasing net of EB channels is also EB. As an example, we construct an injective normal EB channel with an arbitrary outcome operator algebra M acting on an infinite-dimensional separable Hilbert space by using the coherent states and the Bargmann measure. Published by AIP Publishing.