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arXiv:1005.3614 · Physical Review A

High-probability state transfer in spin- 1/2 chains: Analytical and numerical approaches

Edward B. Fel’dman, E. I. Kuznetsova, Alexander Ivanovich Zenchuk

Published Aug 30, 2010• 28 citations

Abstract

This article is devoted to the development of analytical and numerical approaches to the problem of the end-to-end quantum state transfer along the spin-1/2 chain using two methods: (a) a homogeneous spin chain with week end bonds and equal Larmor frequencies and (b) a homogeneous spin chain with end Larmor frequencies different from inner ones. A tridiagonal matrix representation of the $\mathit{XY}$ Hamiltonian with nearest neighbor interactions relevant to the quantum state transfer is exactly diagonalized for a combination of the preceding two methods. In order to take into account interactions of the remote spins, we used numerical simulations of the quantum state transfer in 10-node chains. We compare the state transfer times obtained using the two preceding methods for chains governed by the both $\mathit{XY}$ and $\mathit{XXZ}$ Hamiltonians and using both nearest neighbor and all-node interactions.

DOI: 10.1103/physreva.82.022332

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This article is devoted to the development of analytical and numerical approaches to the problem of the end-to-end quantum state transfer along the spin-1/2 chain using two methods: (a) a homogeneous spin chain with week end bonds and equal Larmor frequencies and (b) a homogeneous spin chain with end Larmor frequencies different from inner ones. A tridiagonal matrix representation of the

\mathit{XY}

Hamiltonian with nearest neighbor interactions relevant to the quantum state transfer is exactly diagonalized for a combination of the preceding two methods. In order to take into account interactions of the remote spins, we used numerical simulations of the quantum state transfer in 10-node chains. We compare the state transfer times obtained using the two preceding methods for chains governed by the both

\mathit{XY}

and

\mathit{XXZ}

Hamiltonians and using both nearest neighbor and all-node interactions.

Abstract We consider the M -neighbor approximation in the problem of one-qubit pure state transfer along the N -node zigzag and alternating spin chains governed by the XXZ -Hamiltonian with the dipole-dipole interaction. We show that always M > 1, i.e., the nearest neighbor approximation is not applicable to such interaction. Moreover, only all-node interaction ( M = N − 1) properly describes the dynamics in the alternating chain. We reveal the region in the parameter space characterizing the chain geometry and orientation which provide the high-probability state-transfer. The optimal state-transfer probability and appropriate time instant for the zigzag and alternating chains are compared.

High-Probability Quantum State Transfer between nodes of an Open Chain of 1/2 Spins with

{XXZ}

Hamiltonian

This paper concerns the problem of the high probability state transfer (HPST) among different nodes of the chains of 1/2-spins with the

{XXZ}

Hamiltonian. This effect is possible due to the suitably distributed weak bonds in the chain. We show that the arbitrary initial state may be transfered with the high probability to any node of the four-node chain with strong end bonds. Varying bonds in the six-node chain one can provide the HPSTs among two end nodes and two symmetrically placed inner nodes. We also construct the eight-node chain with the HPSTs among all of them.

We consider the high-probability state transfers and entanglements between different nodes of the spin-

1/2

chains governed by the

{XXZ}

Hamiltonian using the inhomogeneous stationary external magnetic field. Examples of three-, four-, ten- and twenty-node chains are represented. A variant of realization of the proper inhomogeneous magnetic field is discussed.

In this paper we build on the ideas presented in previous works for perfectly transferring a quantum state between opposite ends of a spin chain using a fixed Hamiltonian. While all previous studies have concentrated on nearest-neighbor couplings, we demonstrate how to incorporate additional terms in the Hamiltonian by solving an inverse eigenvalue problem. We also explore issues relating to the choice of the eigenvalue spectrum of the Hamiltonian, such as the tolerance to errors and the rate of information transfer.

A method for exactly diagonalizing the XY Hamiltonian of an alternating open chain of spins s = 1/2 has been proposed on the basis of the Jordan-Wigner transformation and analysis of the dynamics of spinless fermions. The multiple-quantum spin dynamics of alternating open chains at high temperatures has been analyzed and the intensities of multiple-quantum coherences have been calculated. The problem of the transfer of a quantum state from one end of the alternating chain to the other is studied. It has been shown that the ideal transfer of qubits is possible in alternating chains with a larger number of spins than that in homogeneous chains.