Flag-Based Control of Quantum Purity for $n=2$ Systems

TL;DR

This paper explores a control method for quantum systems that adjusts the system's state purity by manipulating the quantum flag, providing insights into the dynamics of purity in two-level quantum systems.

Contribution

It introduces a novel flag-based control approach for quantum purity, deriving differential equations and identifying critical points in the Bloch sphere for two-level systems.

Findings

Identifies threads of points maximizing and minimizing purity derivative.

Derives differential equations governing purity dynamics.

Maps regions of increasing and decreasing purity inside the Bloch ball.

Abstract

This paper investigates the fast Hamiltonian control of $n=2$ density operators by continuously varying the flag as one moves away from the completely mixed state. In general, the critical points and zeros of the purity derivative can only be solved analytically in the limit of minimal purity. We derive differential equations that maintain these features as the purity increases. In particular, there is a thread of points in the Bloch ball that locally maximizes the purity derivative, and a corresponding thread that minimizes it. Additionally, we show there is a closed surface of points inside of which the purity derivative is positive, and inside of which is negative. We argue that this approach may be useful in studying higher-dimensional systems.