Topology of Entanglement Evolution of Two Qubits

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 ∩ 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.

The Universal Coefficient Theorem and Quantum Field Theory

“…The ideas behind it have their origin far deeper into the past. They have already been mentioned in [294], [295] and several conclusions have been extracted in [296], [297]. The new formulation of the ER-EPR duality basically reminds us that the statistical connection between space-like separated regions of vacuum associated to generic quantum field theories may have a topological interpretation as well.…”

Quantum correlations in a noisy neutron interferometer

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“…The connection between space-time topology and entanglement however remains an unproved conjecture. The ideas behind it have already been mentioned in [5,6,10] and several conclusions have been extracted in [7,8]. The new formulation of the ER-EPR duality basically reminds us that the statistical correlation between space-like separated regions associated to generic quantum field theories may have a topological interpretation as well.…”