Entangling power of symmetric two-qubit quantum gates

TL;DR

This paper analyzes the entangling power of symmetric two-qubit quantum gates, providing a geometric framework for classifying their local equivalence and identifying conditions for maximal entanglement, with applications to various quantum models.

Contribution

It introduces a geometric description of symmetric two-qubit gates using Lie algebra root vectors and characterizes perfect entanglers within this framework.

Findings

One quarter of symmetric two-qubit gates are perfect entanglers.

A geometric method classifies local equivalence classes of gates.

Applications demonstrated on models like Heisenberg and Lipkin-Meshkov-Glick.

Abstract

The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local-invariant. A geometric description of the local equivalence classes of gates is given in terms of the $\mathfrak{su}(3)$ Lie algebra root vectors. These vectors define a primitive cell with hexagonal symmetry on a plane, and through the Weyl group the minimum area on the plane containing the whole set of locally equivalent quantum gates is identified. We give conditions to determine when a given quantum gate produces maximally entangled states from separable ones (perfect entanglers). We found that these gates correspond to one fourth of the whole set of locally-distinct quantum gates. The theory developed here is applicable to three-level systems in general, where the non-locality of a quantum gate is related to its capacity to perform non-rigid transformations on the Majorana constellation of a state. The results are illustrated by an anisotropic Heisenberg model, the Lipkin-Meshkov-Glick model, and two coupled quantized oscillators with cross-Kerr interaction.