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arXiv:0911.1930 · Journal of Physics A Mathematical and Theoretical

Entanglement and magnetic order

Luigi Amico A, Rosario Fazio

Published Dec 2, 2009• 23 citations
arXivHTMLPDF

Abstract

In recent years quantum statistical mechanics have benefited of cultural interchanges with quantum information science. There is much evidence that quantifying the entanglement allows a fine analysis of many relevant properties of many-body quantum systems. Here we review the relation between entanglement and the various types of magnetic order occurring in interacting spin systems.

DOI
10.1088/1751-8113/42/50/504001

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In recent years quantum statistical mechanics have benefited of cultural interchanges with quantum information science. There is much evidence that quantifying the entanglement allows a fine analysis of many relevant properties of many-body quantum systems. Here we review the relation between entanglement and the various types of magnetic order occurring in interacting spin systems.
Abstract We briefly review recent work on entanglement with particular emphasis on its behaviour in interacting spin systems. As already pointed out by several authors a general understanding of the properties of entanglement in condensed matter systems may lead to a deeper comprehension of both dynamics and thermodynamics of many‐body systems. After a brief resume of known means to identify and quantify entanglement, we take interacting spins on a network as an example. We briefly discuss critical properties of entanglement, if the system undergoes a quantum phase transition, and some aspects of its dynamical behaviour.
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Entanglement has been extensively studied from the perspective of Quantum Information Processing for the last decade and various measures of entanglement, to quantify it are now available. Experimentally also it has been possible to measure entanglement in pure state, mostly in optics. However, in recent years there has been significant development in Quantum Condensed Matter systems, especially spin chains, where extraction of entanglement has been made possible through macroscopic witness operators like magnetic susceptibility. In this work, we have studied the magnetic properties of some spin chains, where the exchange interaction can be tuned by preparation technique and we have extracted a magnetic field and temperature dependence of entanglement from susceptibility and magnetization data taken at various temperatures and fields. The spin chains considered here fall in two categories, first one is an alternating dimer model and the second is an infinite spin chain. We have given the experimental plots of entanglement as a function of temperature and magnetic fields and compared them with the theoretical plots.
The review is devoted to two important quantities characterizing many-body systems, order indices and the measure of entanglement production. Order indices describe the type of order distinguishing statistical systems. Contrary to the order parameters characterizing systems in the thermodynamic limit and describing long-range order, the order indices are applicable to finite systems and classify all types of orders, including long-range, mid-range, and short-range orders. The measure of entanglement production quantifies the amount of entanglement produced in a many-partite system by a quantum operation. Despite that the notions of order indices and entanglement production seem to be quite different, there is an intimate relation between them, which is emphasized in the review.
With the development of quantum information science, the active manipulation of quantum systems is becoming an important research frontier. To build realistic quantum information processors, one of the challenges is to implement arbitrary desired operations with high precision on quantum systems. A large number of quantum control methods and relevant numerical techniques have been put forward in recent years, such as quantum optimal control and quantum feedback control. Nuclear magnetic resonance (NMR) spin systems offer an excellent testbed to develop benchmark tools and techniques for controlling quantum systems. In this review paper, we briefly introduce some of the basic control ideas developed for NMR systems in recent years. We first explain, for the liquid spin systems, the physics of various couplings and the causes of relaxation effects. These mechanisms govern the system dynamics, and thus are crucial for constructing rigorous and efficient control models. We also identify three types of available control means: 1) raido-frequency fields as coherent controls; 2) phase cycling, gradient fields and relaxation effects as non-unitary controls; 3) radiation damping effect as feedback control mechanism. Then, we elucidate some important control tasks, which may arise from the conventional NMR spectroscopy (e.g., pulse design and polarization transfer) or from quantum information science (e.g., algorithmic cooling and pseudo-pure state preparation). In the last part, we review some of the most important control methods that are applicable to NMR control tasks. For systems with a relatively small number of spins, it is possible to use analytic optimal control theory to realize the target unitary operations. However, for larger systems, numerical methods are necessary. The gradient ascent pulse engineering algorithm and pulse compiler techniques are the most successful techniques for implementing complicated quantum networks currently. There are some interesting topics of utilizing radiation damping and relaxation effects to achieve more powerful controls. Finally, we give an outline of the possible future work.