Decoherence of a qubit as a diffusion on the Bloch sphere

TL;DR

This paper explores qubit decoherence using geometric quantum mechanics, representing quantum states as probability distributions on the Bloch sphere, and links complete positivity to the ellipticity of a differential operator.

Contribution

It introduces a geometric framework for analyzing qubit decoherence, connecting quantum evolution to differential operators on the Bloch sphere.

Findings

Decoherence corresponds to diffusion processes on the Bloch sphere.

Complete positivity is characterized by the ellipticity of a deformed Laplacian.

The framework provides a geometric interpretation of quantum dynamics.

Abstract

We analyze qubit decoherence in the framework of geometric quantum mechanics. In this framework the qubit density operators are represented by probability distributions which are also the K\"ahler functions on the Bloch sphere. Interestingly, the complete positivity of the quantum evolution is recovered as ellipticity of the second order differential operator (deformed Laplacian) which governs the evolution of the probability distribution.