Information propagation through disordered quantum spin chains

Abstract

One of the central aims of quantum information theory is to exploit quantum mechanical phenomena such as superposition and entanglement to build a quantum computer — a device capable of effciently performing computational tasks which are not feasible on a classical computer. In order to achieve this goal it will be necessary to connect together the different components of a quantum computer with channels which are capable of quickly and faithfully transmitting quantum states. It has been suggested that a good channel for this purpose is a chain of permanently coupled quantum spins and we begin by reviewing some of the protocols which have been previously developed to transfer quantum information along such a spin chain. We then examine the effects of noise on the propagation of quantum information along spin chains, using a toolkit of methods which include Lieb-Robinson bounds, the Jordan-Wigner transform and correlation functions. Several fundamentally different noise models are considered including a static (time-independent) on-site disorder model and two different models of fluctuating (time-dependent) on-site disorder. Each noise model has a different and distinctive effect on the propagation of information — static disorder leads to exponential localisation whilst dynamic disorder can lead to diffusive or even ballistic propagation of information. We finish by reviewing how a spin chain (whether noisy or noise-free) can be viewed as a depolarising channel and we make a detailed study of the geometrical structure of all possible depolarising channels with respect to various bases, concentrating on the Pauli, Gell-Mann and Heisenberg-Weyl bases. In particular, we show precisely when the set of all possible depolarising channels forms a simplex in compression space.