Entanglement-breaking channels with general outcome operator algebras

Abstract

A unit-preserving and completely positive linear map, or a channel, $\Lambda \colon \mathcal{A} \to \mathcal{A}_{\mathrm{in}}$ between $C^\ast$-algebras $\mathcal{A}$ and $\mathcal{A}_{\mathrm{in}}$ is called entanglement-breaking (EB) if $\omega \circ( \Lambda \otimes \mathrm{id}_{\mathcal{B}} ) $ is a separable state for any $C^\ast$-algebra $\mathcal{B}$ and any state $\omega$ on the injective $C^\ast$-tensor product $\mathcal{A}_{\mathrm{in}} \otimes \mathcal{B} .$ 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 $C^\ast$-algebra, generalizing 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 $2 \leq n < \infty ;$ (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.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 $\mathcal{M}$ acting on an infinite-dimensional separable Hilbert space by using the coherent states and the Bargmann measure.