A unit-preserving and completely positive linear map, or a channel, Λ:A→Ain between C∗-algebras A and Ain is called entanglement-breaking (EB) if ω' - ' (Λ ' - idB) is a separable state for any C∗-algebra B and any state ω on the injective C∗-tensor product Ain' - B. In this paper, we establish the equivalence of the following conditions for a channel Λ with a quantum input space and with a general outcome C∗-algebra, generalizing the known results in finite dimensions: (i) Λ is EB; (ii) Λ has a measurement-prepare form (Holevo form); (iii) n copies of Λ are compatible for all 2 ≤ n < ∞; (iv) countably infinite copies of Λ 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.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.
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