Entanglement-breaking channels with general outcome operator algebras

Abstract

A unit-preserving and completely positive linear map, or a channel, Λ :is a separable state for any C * -algebra B and any state ω on the injective C * -tensor product A in ⊗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 known results in finite dimensions: (i) Λ is EB; (ii) Λ has a measurementprepare 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.

Jordan products of quantum channels and their compatibility

“…The task of determining compatibility of quantum channels, which has been studied recently (see, e.g., [28][29][30][31][32]) can be stated as follows. Given two quantum channels, Φ 1 from X to Y 1 and Φ 2 from X to Y 2 , one determines if there exists another channel Φ from X to Y 1 ⊗ Y 2 that satisfies…”

“…Compatibility of channels has also been investigated recently (see, e.g., [28][29][30][31][32]). Note that two compatible channels do not necessarily possess a unique compatibilizer.…”