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arXiv · 2006.07076

$C^*$-extreme points of positive operator valued measures and unital completely positive maps

Tathagata Banerjee·B V Rajarama Bhat·Manish Kumar
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Abstract

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 ${\mathbb C}^n$) is $C^*$-extreme in the set of unital completely positive maps if and only if it is a unital $*$-homomorphism.

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Tathagata Banerjee, B V Rajarama Bhat, Manish Kumar. 2020-06-12. $C^*$-extreme points of positive operator valued measures and unital completely positive maps. https://doi.org/10.1007/s00220-021-04245-1

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