On quantum channels generated by covariant positive operator-valued measures on a locally compact group

TL;DR

This paper introduces a new class of quantum channels generated by covariant POVMs linked to locally compact groups, utilizing Pontryagin duality to establish isometrical isomorphisms, and explores their properties including a complementary channel.

Contribution

It presents a novel construction of quantum channels from covariant POVMs on locally compact groups using Pontryagin duality, and analyzes their structure and relationships.

Findings

Established an isometrical isomorphism between Hilbert-Schmidt operators and $L^2(\hat G imes G)$

Defined a pair of hybrid quantum channels from covariant POVMs, including a measurement and a complementary channel

Demonstrated the properties and relationships of these channels in the context of group representations

Abstract

We introduce positive operator-valued measure (POVM) generated by the projective unitary representation of a direct product of locally compact Abelian group $G$ with its dual $\hat G$. The method is based upon the Pontryagin duality allowing to establish an isometrical isomorphism between the space of Hilbert-Schmidt operators in $L^2(G)$ and the Hilbert space $L^2(\hat G\times G)$. Any such a measure determines a pair of hybrid (containing classical and quantum parts) quantum channels consisting of the measurement channel and the channel transmitting an initial quantum state to the ensemble of quantum states on the group. It is shown that the second channel can be called a complementary channel to the measurement channel.