Extremal generalized quantum measurements

TL;DR

This paper characterizes extremal measurements on sections of quantum states, including quantum channels, providing explicit conditions for extremality and illustrating with qubit examples.

Contribution

It introduces extremality conditions for measurements on sections of quantum states and characterizes generalized POVMs, especially in the context of quantum networks and channels.

Findings

Derived extremality conditions for measurements on sections of quantum states.

Characterized extremal generalized POVMs and applied results to quantum channels.

Provided explicit extremality conditions for qubit channel measurements.

Abstract

A measurement on a section K of the set of states of a finite dimensional C*-algebra is defined as an affine map from K to a probability simplex. Special cases of such sections are used in description of quantum networks, in particular quantum channels. Measurements on a section correspond to equivalence classes of so-called generalized POVMs, which are called quantum testers in the case of networks. We find extremality conditions for measurements on K and characterize generalized POVMs such that the corresponding measurement is extremal. These results are applied to the set of channels. We find explicit extremality conditions for two outcome measurements on qubit channels and give an example of an extremal qubit 1-tester such that the corresponding measurement is not extremal.