Characterizing the Depolarizing Quantum Channel in Terms of Riemannian Geometry

TL;DR

This paper uses Riemannian geometry to analyze how depolarizing channels affect quantum states, revealing geometric insights into state contraction and randomization in the Bloch sphere representation.

Contribution

It introduces a geometric framework to characterize depolarizing channels using Riemannian tools based on quantum state distinguishability.

Findings

Depolarizing channels cause a contraction of the Bloch sphere geometry.

Randomization is represented by a smaller quantum line element.

Deformed geodesics reflect the contraction of state components.

Abstract

We explore the conceptual usefulness of Riemannian geometric tools induced by the statistical concept of distinguishability in quantifying the effect of a depolarizing channel on quantum states. Specifically, we compare the geometries of the interior of undeformed and deformed Bloch spheres related to density operators on a two-dimensional Hilbert space. We show that randomization emerges geometrically through a smaller infinitesimal quantum line element on the deformed Bloch sphere while the uniform contraction manifests itself via a deformed set of geodesics where the spacial components of the deformed four-Bloch vector are simply the contracted versions of the undeformed Bloch vector components.