Geometric Aspects of Entanglement Generating Hamiltonian Evolutions

TL;DR

This paper explores the geometric properties of Hamiltonian evolutions that generate entanglement in two-qubit systems, linking geometric efficiency with entanglement measures and identifying characteristics of time-optimal trajectories.

Contribution

It introduces a geometric framework for analyzing entanglement evolution, revealing that time-optimal paths are efficient, curvature-free, and involve less entanglement, with distinctions based on state orthogonality.

Findings

Time-optimal evolutions have high geodesic efficiency and zero curvature.

Suboptimal trajectories between orthogonal states involve longer paths and more energy wastage.

Higher initial nonlocality facilitates reaching maximal entanglement from separable states.

Abstract

We examine the pertinent geometric characteristics of entanglement that arise from stationary Hamiltonian evolutions transitioning from separable to maximally entangled two-qubit quantum states. From a geometric perspective, each evolution is characterized by means of geodesic efficiency, speed efficiency, and curvature coefficient. Conversely, from the standpoint of entanglement, these evolutions are quantified using various metrics, such as concurrence, entanglement power, and entangling capability. Overall, our findings indicate that time-optimal evolution trajectories are marked by high geodesic efficiency, with no energy resource wastage, no curvature (i.e., zero bending), and an average path entanglement that is less than that observed in time-suboptimal evolutions. Additionally, when analyzing separable-to-maximally entangled evolutions between nonorthogonal states, time-optimal evolutions demonstrate a greater short-time degree of nonlocality compared to time-suboptimal evolutions between the same initial and final states. Interestingly, the reverse is generally true for separable-to-maximally entangled evolutions involving orthogonal states. Our investigation suggests that this phenomenon arises because suboptimal trajectories between orthogonal states are characterized by longer path lengths with smaller curvature, which are traversed with a higher energy resource wastage compared to suboptimal trajectories between nonorthogonal states. Consequently, a higher initial degree of nonlocality in the unitary time propagators appears to be essential for achieving the maximally entangled state from a separable state. Furthermore, when assessing optimal and suboptimal evolutions...