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arXiv:0809.4833 · Physical Review A

Information propagation through quantum chains with fluctuating disorder

Christian K. Burrell, Jens Eisert, Tobias J. Osborne

Published Nov 12, 2009• 50 citations
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Abstract

We investigate the propagation of information through one-dimensional nearest-neighbor interacting quantum spin chains in the presence of external fields which fluctuate independently on each site. We study two fundamentally different models: (i) a model with general nearest-neighbor interactions in a field which fluctuates in both strength and direction and (ii) the $XX$ chain placed in a fluctuating field aligned in the $z$ direction. In both cases we find that information propagation is suppressed in a way which is quite different from the suppression observed when the $XX$ model is placed in a statically disordered field.

DOI
10.1103/physreva.80.052319

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We investigate the propagation of information through one-dimensional nearest-neighbor interacting quantum spin chains in the presence of external fields which fluctuate independently on each site. We study two fundamentally different models: (i) a model with general nearest-neighbor interactions in a field which fluctuates in both strength and direction and (ii) the XXXXXX chain placed in a fluctuating field aligned in the zzz direction. In both cases we find that information propagation is suppressed in a way which is quite different from the suppression observed when the XXXXXX model is placed in a statically disordered field.
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.
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We study the magnetic properties of two types of one-dimensional XXXXXX spin-12\frac{1}{2}21​ chains. The first type has only nearest-neighbor interactions which can be either antiferromagnetic or ferromagnetic, and the second type has both nearest-neighbor and next-nearest-neighbor interactions, but only antiferromagnetic in character. We study these systems in the presence of low magnetic fields both analytically and numerically. Comparison of results shows a close relation between the two systems, which is in agreement with results previously found in Heisenberg chains by means of a numerical real-space renormalization-group procedure.
We study the dynamics of entanglement for a one-dimensional spin system, where spins are coupled through nearest-neighbor exchange interaction and subject to different external magnetic fields. First we examine the system size effect on the entanglement with three different external magnetic fields changing with time ttt: an exponential function ${e}^{[\ensuremath{-}Kt]}$ and two periodic sin[Kt]\mathrm{sin}[Kt]sin[Kt] and cos[Kt]\mathrm{cos}[Kt]cos[Kt] functions, where KKK is a control parameter. We have found that the entanglement fluctuates shortly after a disturbance by an external magnetic field when the system size is small. For larger system size, the entanglement reaches a stable state for a long time before it fluctuates. However, this fluctuation of entanglement disappears when the system size goes to infinity. We also show that in a periodic external magnetic field, the nearest-neighbor entanglement displays a periodic structure with a period related to that of the magnetic field. Moreover, changing the direction of the magnetic field will destroy the concurrence in the system.
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