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arXiv:1302.0366

Transfers of entanglement qudit states in quantum networks

Marek Sawerwain, Joanna Wiśniewska

Published Feb 2, 2013• 0 citations
arXivHTMLPDF

Abstract

The issue of quantum states' transfer -- in particular, for so-called Perfect State Transfer (PST) -- in the networks represented by the spin chains seems to be one of the major concerns in quantum computing. Especially, in the context of future communication methods that can be used in broadly defined computer science. The chapter presents a definition of Hamiltonian describing the dynamics of quantum data transfer in one-dimensional spin chain, which is able to transfer the state of unknown qudits. The main part of the chapter is the discussion about possibility of entangled states' perfect transfer, in particular, for the generalized Bell states for qudits. One of the sections also contains the results of numerical experiments for the transmission of quantum entangled state in a noisy quantum channel.

DOI
10.48550/arxiv.1302.0366

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The issue of quantum states' transfer -- in particular, for so-called Perfect State Transfer (PST) -- in the networks represented by the spin chains seems to be one of the major concerns in quantum computing. Especially, in the context of future communication methods that can be used in broadly defined computer science. The chapter presents a definition of Hamiltonian describing the dynamics of quantum data transfer in one-dimensional spin chain, which is able to transfer the state of unknown qudits. The main part of the chapter is the discussion about possibility of entangled states' perfect transfer, in particular, for the generalized Bell states for qudits. One of the sections also contains the results of numerical experiments for the transmission of quantum entangled state in a noisy quantum channel.
Following the prescription of Ref. \cite{PST} in which perfect state transference (PST) of a qubit over distance regular spin networks was discussed, in this paper PST of an arbitrary ddd-level quantum state (qudit) over antipodes of more general networks called pseudo distance-regular networks, is investigated. In fact, the spectral analysis techniques used in the previous work \cite{PST}, and algebraic structures of pseudo distance-regular graphs are employed to give an explicit formula for suitable coupling constants in the Hamiltonians so that the state of a particular qudit initially encoded on one site will evolve freely to the opposite site without any dynamical control, i.e., we show that how to derive the parameters of the system so that PST can be achieved. Keywords:Perfect state transfer, ddd-level quantum state, Stratification, Pseudo-distance-regular network PACs Index: 01.55.+b, 02.10.Yn
We propose a class of qubit networks that admit perfect state transfer of any two-dimensional quantum state in a fixed period of time. We further show that such networks can distribute arbitrary entangled states between two distant parties, and can, by using such systems in parallel, transmit the higher-dimensional systems states across the network. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to NNN-qubit spin networks of identical qubit couplings, we show that 2log3N2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{3}N2log3​N is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits then perfect state transfer can be achieved over arbitrarily long distances in a linear chain. This paper expands and extends the work done by Christandl et al., Phys. Rev. Lett. 92, 187902 (2004).
Christandl et al. have noted that the ddd-dimensional hypercube can be projected to a linear chain with d+1d+1d+1 sites so that, by considering fixed but different couplings between the qubits assigned to the sites, the perfect state transfer (PST) can be achieved over arbitrarily long distances in the chain [Phys. Rev. Lett. 92, 187902 (2004); Phys. Rev. A 71, 032312 (2005)]. In this work we consider distance-regular graphs as spin networks and note that any such network (not just the hypercube) can be projected to a linear chain and so can allow PST over long distances. We consider some particular spin Hamiltonians which are the extended version of those of Christandl et al. Then, by using techniques such as stratification of distance-regular graphs and spectral analysis methods, we give a procedure for finding a set of coupling constants in the Hamiltonians so that a particular state initially encoded on one site will evolve freely to the opposite site without any dynamical control, i.e., we show how to derive the parameters of the system so that PST can be achieved. It is seen that PST is only allowed in distance-regular spin networks for which, starting from an arbitrary vertex as reference vertex (prepared in the initial state which we wish to transfer), the last stratum of the networks with respect to the reference state contains only one vertex; i.e., stratification of these networks plays an important role which determines in which kinds of networks and between which vertices of them, PST can be allowed. As examples, the cycle network with even number of vertices and ddd-dimensional hypercube are considered in details and the method is applied for some important distance-regular networks.
We investigate the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. We consider first the simplest possible spin chain, where the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are constant throughout the chain. The time evolution of a single spin state is determined, and this time evolution is illustrated by means of an animation. Some years ago it was discovered that when the spin chain data are of a special form so-called perfect state transfer takes place. These special spin chain data can be linked to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials. We discuss here the case related to Krawtchouk polynomials, and illustrate the possibility of perfect state transfer by an animation showing the time evolution of the spin chain from an initial single spin state. Very recently, these ideas were extended to discrete orthogonal polynomials of q -hypergeometric type. Here, a remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q -Krawtchouk polynomials. This case is discussed here, and again illustrated by means of an animation.
We propose a class of qubit networks that admit the perfect state transfer of any quantum state in a fixed period of time. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to N-qubit spin networks of identical qubit couplings, we show that 2log3N is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits, then perfect state transfer can be achieved over arbitrarily long distances in a linear chain.
In quantum computation and quantum information processing, the manipulation and engineering of quantum systems to suit certain purposes are an ongoing task. One such example is quantum state transfer (QST), an essential requirement for both quantum communication and large-scale quantum computation. Here we engineer a chain of four superconducting qubits with tunable couplers to realize the perfect state transfer (PST) protocol originally proposed in quantum spin networks and successfully demonstrate the efficient transfer of an arbitrary single-qubit state from one end of the chain to the other, achieving a high fidelity of 0.986 in just 25 ns. This demonstrated QST is readily to extend to larger chain and multi-node configurations, thus serving as a desirable tool for scalable quantum information processing.