C^∗ -Extreme Points of Positive Operator Valued Measures and Unital Completely Positive Maps

Authors

Tathagata Banerjee, Indian Statistical Institute Bangalore
B. V.Rajarama Bhat, Indian Statistical Institute Bangalore
Manish Kumar, Indian Statistical Institute Bangalore

Article Type

Research Article

Publication Title

Communications in Mathematical Physics

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 Cn) is C∗-extreme in the set of unital completely positive maps if and only if it is a unital ∗ -homomorphism.

First Page

1235

Last Page

1280

DOI

10.1007/s00220-021-04245-1

Publication Date

12-1-2021

Comments

Open Access, Green

Recommended Citation

Banerjee, Tathagata; Bhat, B. V.Rajarama; and Kumar, Manish, "C^∗ -Extreme Points of Positive Operator Valued Measures and Unital Completely Positive Maps" (2021). Journal Articles. 1672.
https://digitalcommons.isical.ac.in/journal-articles/1672