Topology of Entanglement Evolution of Two Qubits

TL;DR

This paper classifies the different ways entanglement can evolve in a two-qubit system, using a geometric approach based on the polarization vector and the topology of separable states.

Contribution

It provides a rigorous topological framework for understanding entanglement evolution, including a classification of four distinct behaviors and a method to analyze specific models.

Findings

Identifies four types of entanglement evolution.

Introduces the concept of distance-Markovian dynamics.

Provides a method to compute critical points in entanglement evolution.

Abstract

The dynamics of a two-qubit system is considered with the aim of a general categorization of the different ways in which entanglement can disappear in the course of the evolution, e.g., entanglement sudden death. The dynamics is described by the function ~n(t), where ~n is the 15-dimensional polarization vector. This representation is particularly useful because the components of ~n are direct physical observables, there is a meaningful notion of orthogonality, and the concurrence C can be computed for any point in the space. We analyze the topology of the space S of separable states (those having C = 0), and the often lower-dimensional linear dynamical subspace D that is characteristic of a specific physical model. This allows us to give a rigorous characterization of the four possible kinds of entanglement evolution. Which evolution is realized depends on the dimensionality of D and of D \cap S, the position of the asymptotic point of the evolution, and whether or not the evolution is "distance-Markovian", a notion we define. We give several examples to illustrate the general principles, and to give a method to compute critical points. We construct a model that shows all four behaviors.