Controllability of Symmetric Spin Networks

TL;DR

This paper investigates the controllability of symmetric spin networks with Ising interactions, showing that permutation-invariant evolutions are controllable and providing explicit control laws for low-dimensional cases, including entangled states like GHZ and W states.

Contribution

It demonstrates controllability within permutation-invariant subspaces of symmetric spin networks and offers explicit control strategies for low-dimensional cases.

Findings

Controllability holds for permutation-invariant evolutions.

Explicit control laws are provided for n=2 and n=3.

Permutation-invariant states include important entangled states like GHZ and W.

Abstract

We consider a network of n spin 1/2 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n=2 and n=3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between any two permutation invariant states. This class of states includes highly entangled states such as GHZ states and W states, which are of interest in quantum information.