Communications in Mathematical Physics
We study the quantum (C∗) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, C∗-extreme points of normalized POVMs on countable spaces (in particular for finite sets) are always spectral measures (normalized projection valued measures). More generally it is shown that atomic C∗-extreme points are spectral. A Krein–Milman type theorem for POVMs has also been proved. As an application it is shown that a map on any commutative unital C∗-algebra with countable spectrum (in particular Cn) is C∗-extreme in the set of unital completely positive maps if and only if it is a unital ∗ -homomorphism.
Banerjee, Tathagata; Bhat, B. V.Rajarama; and Kumar, Manish, "C∗-Extreme Points of Positive Operator Valued Measures and Unital Completely Positive Maps" (2021). ISI Best Publications. 4.