Optimal state transfer in long-range interacting spin-chains

In recent years, the interest in quantum communication channels has experienced a rapid increase, also in view of their use as a bus between registers and processor within a quantum computer. Despite the fact that the majority of protocols for quantum communication rely on photons, because of their weak interaction with the environment, and of the welldeveloped optical fiber technology, it is not always possible to use photons when one needs a frequent exchange of information between distant qubits. For example, in the case we have mentioned, the communication between different parts of a quantum computer would require a continous conversion of stationary qubits (i.e. the information stored in the components of the quantum computer) into ying qubits (i.e. photons), in order to transmit information. This procedure leads to several interfacing problems between the two different kinds of physical systems, that could be avoided by using, as a quantum channel, the same kind of physical system that is used for realizing the quantum computer. Indeed, a seminal paper by Bose, suggested to use spin-chains as a quantum channel for short or mid-range communication, showing that, by means of the magnetic interaction between the spins composing the chain, the information is transferred by only letting the system evolve dynamically, without the requirement of any external control. Transferring quantum information between distant qubits through spin-chains would be highly desirable, as, in general, this is a procedure which requires the repeated application of swapping gates and is, consequently, higly experimentally demanding. After the first proposal, there has been a spread of works investigating the dynamical behaviour of spinchains and possible optimization techniques for enhancing its performances as a quantum channel. There has been proposed a large amount of schemes for reaching perfect state transfer by optimising the system over a variety of parameters, as, for example, the robustness against errors and a restricted ability to engineer the state, or, on the opposite, the ability of engineering the couplings and a definite interval of time for the state transfer, or by allowing other kinds of control over the chain. However, all the schemes proposed, not only present some kind of side-effects, as, for example, a potentially indefinite waiting time for observing the transfer, or a difficult generalization to the transfer of multiple-qubit states, or are not scalable with the size of the system, or require a demanding technology in order to be implemented, but also rely on the nearest-neighbour modellization of the chain. The nearest-neighbour model appears to be a rather theoretical model, with respect to a long-range interacting model, such as a spin-chain which exhibits dipolar couplings. In fact, the experimental implementations of spin-chain based on trapped particles, such as ions or electrons, exhibit dipolar couplings. Indeed, trapped particles, like ions or electrons, are suitable for implementing both a scalable quantum processor, and a quantum channel, so they represent one the most interesting playgrounds for the implementation of spin-chain. In literature, there have been only few examples of works taking into account long-range interactions, which, in general, have always been regarded as a sort of perturbation with respect to the nearest-neighbour (NN) model. However there are relevant features of the long-range interactions that cannot be captured within the frame of only NN-coupling. An example is provided by the completely different behaviour in presence of defects such as vacancies along the chain. The presence of empty sites, in fact, in the NN case would prevent the use of the system as a quantum channel, whereas the long-range interacting chain is more robust against this kind of errors, which, for example, could be the consequence of an imperfect chain-filling. We start our investigation with the aim of reaching a deeper understanding of the dynamics of long-range interacting systems, and of proposing an optimization scheme, for this class of chains, which could avoid, as much as possible, the drawbacks of the previous proposals. Our goal is to find a general and easy-to-implement prescription that allows to significanlty enhance the performances of the channel. In order to accomplish this task we adopt an approach which lead us to start our investigation from the very beginning. In fact, we look for the most general considerations about system properties, like symmetry, energy spectrum and eigenvectors, that allows a generic physical system to transfer perfectly information, in order to understand their role in the communiation process. We find that, given a fixed transmission distance, perfect state transfer, moreover in the shortest possible time, can be achieved only by a system made of two spins. This ideal system, however, results unpractical, due to the decrease of the interactions with the inter-spin distance , which makes it impossible to transfer a state over an arbitrarily long distance. In fact, in presence of whatever kind of noise, the coupling between sender and receiver would result too weak after a short distance. The obvious way to circumvent this problem relies on filling the space in between with other spins, as in the ordinary spin-chains. However, this procedure on one side enforces the communication, but on the other, lowers the fidelity and increases the transfer time. What is, then, the smartest way to fill in the space between sender and receiver in order to keep the fidelity of transmission close to unity and the transfer time reasonably short? We find an answer to this question which leads to optimal results and, most notably, largely independent of the number of spins in the chain, hence scal-able. Therefore, our scheme can be applied also to mid-range communication in a rather straightforward way. The key element is a joudicious balance between the ideal two-spin chain and the complete chain with equally spaced spins. We do not resort to specific design or challenging system engineering, but simply identify a general and easy prescription to optimize the performances of the spin chain as a quantum channel. Furthermore, our scheme leads to optimal state transfer also in all the higher excitation subspaces, allowing the transmission of multiple-qubit states and of multiple-entangled states. We show explictly how this applies in the case of two excitations travelling along the chain. The Hamiltonian of the system, indeed, acts on the different excitations as a whole entity, and, in this terms, allows a mapping in which the degrees of freedom relative to the positions of the excitations are mapped into a single index spanning all the states in the configuration space of the system. We study the double excitation case also in connection with the eventuality of thermal noise in the system. This investigation is quite general and could be used also as a scheme appliabe to other situations, such as memory effects in the channel. Our work is organised as follow: in Chap.1 we introduce the basic concepts concerning the quantum channels and introduce the basic protocol of communication through a spin chain that has been originally proposed in Ref. [2]. We also provide an overview of the optimization schemes proposed in literature, and of both their advantages and their disadvantages. In Chap. 2 we illustrate the three main proposals for experimentally implementing the spin-chain dynamics, and show how it naturally arises, in the two most interesting cases, the dipolar-coupling Hamiltonian. After a very short overview of the state of the art concerning long-range interactions, and of the standard way to approach these systems, in Chap.3, we start our original investigation, by examining the most general conditions under which the transfer fidelity is maximized. We analize the role of spatial symmetry in connection with the properties of the eigenvectors and of the relative eigenenergies. These considerations lead us to consider symmetry a necessary condition for perfect state transfer, and to derive also a sufficient condition. We also prove that the ring of qubits is not a suitable configuration for transferring a state between two parties, by showing how it does not match the conditions we have found for perfect state transfer. Furthermore, by means of the symmetry, we provide an explanation for the dependence, of the ring performances, on the parity of the number of spins, despite its ferromagnetic nature. In Chap. 4 we further explore the implications of our results by including defects, e.g. empty sites, in the chain. In particular we first examine a single-hole chain and then skip to the double-hole case. The latter ones lies at the heart of our procedure for achieving, by practical means, perfect state transfer. We show that, within our scheme, not only the performances of the system are dramatically enhanced, but also, given a fixed transmission distance, are invariant under system rescaling. We also extend our procedure to the double-excitation subspace. Finally, we examine the case in which a second excitation is induced in the chain by thermal noise.