Completely positive module maps and completely positive extreme maps

Abstract

Abstract. Let A, B be unital C * -algebras and P∞(A, B) be the set of all completely positive linear maps of A into B. In this article we characterize the extreme elements in P∞ (A, B, p) , p = Φ(1) for all Φ ∈ P∞(A, B, p), and pure elements in P∞(A, B) in terms of a self-dual Hilbert module structure induced by each Φ in P∞(A, B). Let P∞(B(H)) R be the subset of P∞(B(H), B(H)) consisting of R-module maps for a von Neumann algebra R ⊆ B(H).We characterize normal elements in P∞(B(H)) R to be extreme. Results here generalize various earlier results by Choi, Paschke and Lin.

When do pieces determine the whole? Extreme marginals of a completely positive map

“…is a matrix representation of a bounded operator on K. We may rephrase the extremality condition of channels: A channel E ∈ C(K; H) with a minimal set {K j } j of Kraus operators is extremal in C(K; H) if and only if the set {K * j K k } j,k is strongly independent. This result can be found in [11]. • Instruments: A complete characterization of extremal instruments is presented in [7].…”

Non-Gaussian operations on bosonic modes of light: Photon-added Gaussian channels

“…The conjecture was shown to hold true for the case n = 0, i.e., the Gaussian channel case, in Ref. [55] using a theorem by Choi [73,74] that tells us that a channel Φ with Kraus operators {F ℓ [Φ]} ℓ is extremal if and only if the set of associated matrices {W [Φ] ℓℓ ′ } ℓ,ℓ ′ defined as…”

On the extreme points of quantum channels

“…We reprove and extend some of the results in [2,12,11]. Some related results are discussed in [1,14,15,4]. One of the novel features of this paper is the use of the notions and results of complex and real algebraic geometry, and semi-algebraic geometry.…”