Solid-state multiple quantum NMR in quantum information processing: exactly solvable models

0 Institute of Problems of Chemical Physics , RAS, Chernogolovka, Moscow Region 142432 , Russia Multiple quantum (MQ) NMR is an effective tool for the generation of a large cluster of correlated particles, which, in turn, represent a basis for quantum information processing devices. Studying the available exactly solvable models clarifies many aspects of the quantum information. In this study, we consider two exactly solvable models in the MQ NMR experiment: (i) the isolated system of two spin- 12 particles (dimers) and (ii) the large system of equivalent spin- 12 particles in a nanopore. The former model is used to describe the quantum correlations and their relations with the MQ NMR coherences, whereas the latter helps one to model the creation and decay of large clusters of correlated particles. 1. Introduction At present, the perspectives of technological development are incredible without combining classical and quantum devices, i.e. devices whose operation is based on the principles of quantum mechanics. The development of quantum information technology stimulates an in-depth study of the properties of quantum correlations inherent in a quantum system. In particular, there is a problem of identification of those quantum correlations that are responsible for the advantages of quantum computations in comparison with classical counterparts. Entanglement [15], which was originally taken as a measure of such correlations, seemed to not cover all of them. As a consequence, there are quantum systems without entanglement, which, nevertheless, reveal either a quantum non-locality [68] or speed up certain calculations in comparison with the classical analogues [913]. Such observations cause a new stimulus for studying those quantum correlations that are not captured by entanglement. Thus, the concept of quantum discord has been intensively developed during past years [1416]. Originally, the quantum discord was introduced to characterize the impact of classical measurements on a quantum system with the purpose to get the maximal information about this system with the minimal influence on it [17]. At first glance, the quantum discord seems to cover all quantum correlations. However, it was shown that the quantum discord may be either bigger or smaller than the entanglement [15,18,19]. Thus, we may state that the quantum discord involves different quantum correlations than entanglement does, in general, which causes doubts whether the discord captures all correlations. Moreover, the discord is not symmetrical with respect to the subsystem chosen for the projective measurements [14], which stimulates further study of this ambiguity [20] and suggests introducing the discord under two-side projective measurements [21] (a symmetrical discord). Quantum cryptography [22,23], teleportation [2428] and metrology [2931] seem to be most progressive by way of application of quantum systems. Another promising advantage of quantumness, which still is not realized, is the quantum computation. The first result in this direction was demonstrated in Knill et al. [31], where ensemble quantum computing with liquid state nuclear magnetic resonance (NMR) offers the seven-qubit quantum register. However, perspectives of quantum information processing (QIP) based on liquid state NMR cause doubts because of two basic reasons: (i) the number of correlated qubits is very restricted (about 10) [32] and (ii) the quantum correlations in liquid qubit system are not strong. In particular, the entanglement is almost absent [33]. However, quantum correlations are believed to be the main resource of quantum devices allowing, in particular, the speeding up of quantum processes in comparison with their classical analogues. For this reason, development of QIP based on NMR in solids seems to be much more promising, because (i) the number of correlated qubits may be much bigger than in liquids and (ii) there are entangled states in solid quantum systems that must provide advantages for such QIP. The magnetic resonance force microscopy work of Yamamoto et al. [3437] has inspired us to ask Could we design a solid-state NMR QIP device with quasione-dimensional hydroxyapatite or one of its modified derivatives? [38,39]. Our efforts to answer this question have been strongly influenced by NMR work on this material from the 1960s to today [37,4042]. While we realize that other solid-state NMR QIP device proposals exist with modest numbers of qubits [34 37], the unique combination of circumstances suggests that in a macroscopic sample of hydroxyapatite, the ultimate physical limit of 100 qubits could be achievable, in principle, with bulk NMR methods. Physical estimations with current technology indicate that hundreds to thousands of qubits should be within reach. As an example [39], we consider a crystalline sample of calcium hydroxyapatite, Ca5(PO4)3OH (3.5 9.5 9.5 cm) that contains approximately 1024 hydrogens. The microscopic structure of such a sample consists of one-dimensional chains of hydrogens, from hydroxyl groups, with a lattice spacing of 3.44 , and with each chain surrounded by six nearest-neighbour (NN) parallel chains at a distance of 9.42 ; calcium and phosphate ions are interspersed among the chains [38]. When the hydrogen chains are parallel to a strong Zeeman magnetic field along the z direction, the sample has 108 planes oriented perpendicularly to the field, with 1016 hydrogen nuclear spins in each plane. Thus, by adding a static magnetic field gradient along the z -direction, each plane of spins would be identified with a different resonant Zeeman frequency. These individual planes would be the working qubits. Multiple quantum (MQ) NMR spectroscopy provides some insights into the space-filling properties of condensed matter on microscopic and possibly mesoscopic length scales [43]. The interplay of MQ NMR spin dynamics and the dimensionality of the space embedding the spins has been probed in materials with quasi-one-dimensional distributions of spins 12 by Yesinowksi et al. [40,44,45]. Efforts to understand these experimental results were limited to arguments involving less than typically 10 spins. An exact solution for MQ NMR spin dynamics in an infinite one-dimensional quantum spin chain with NN dipolar interactions has been obtained in the high-temperature approximation [46] as well as in the low-temperature case [47]. This model is the first exactly solvable model in MQ NMR for a system with a macroscopic number of coupled spins. Numerous reasons exist to study simplified and tractable one-dimensional models [48]. Our motivations and justifications include the following: (i) the hope to gain insights, understanding and predictions of some features of MQ NMR in materials with quasi-one-dimensional distributions of spins [40,44,45], (ii) in many cases, consideration of only NN dipolar interactions in a one-dimensional chain is a reasonable approximation because 83 per cent of the avera (...truncated)