Geometry of phase-covariant qubit channels

TL;DR

This paper explores the geometric structure of phase-covariant qubit channels, deriving volume measures and shapes for various subclasses, including entanglement-breaking and time-local generator channels.

Contribution

It provides an analytical geometric analysis of non-unital phase-covariant qubit channels using Choi-Jamiołkowski states and computes their volumes and shapes.

Findings

Derived Hilbert-Schmidt line and volume elements for these channels.

Computed the shapes and volumes of phase-covariant channels.

Analyzed entanglement-breaking and time-local generator channels.

Abstract

We analyze the geometry on the space of non-unital phase-covariant qubit maps. Using the corresponding Choi-Jamio{\l}kowski states, we derive the Hilbert-Schmidt line and volume elements using the channel eigenvalues together with the parameter that characterizes non-unitality. We find the shapes and analytically compute the volumes of phase-covariant channels, in particular entanglement breaking and obtainable with time-local generators.