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High-Probability Quantum State Transfer between nodes of an Open Chain of 1/2 Spins with ${XXZ}$ Hamiltonian

Edward B. Fel’dman, Alexander Ivanovich Zenchuk

Published Sep 11, 2008• 0 citations
OpenAlex

Abstract

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.

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High-Probability Quantum State Transfer between nodes of an Open Chain of 1/2 Spins with XXZ{XXZ}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{XXZ}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-1/21/21/2 chains governed by the XXZ\mathit{XXZ}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.
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 XY\mathit{XY}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 XY\mathit{XY}XY and XXZ\mathit{XXZ}XXZ Hamiltonians and using both nearest neighbor and all-node interactions.
The ability to transfer quantum information from one location to another with high probability is of central importance to quantum information science. Unfortunately, for the simplest system of a uniform chain (a spin chain or a particle in a one-dimensional lattice), the state transfer time grows exponentially in the chain length NNN at fixed transfer probability. In this work we show that the addition of an impurity near each end point, coupled to the uniform chain with strength www, is sufficient to ensure efficient and high-probability state transfer. An eigenstate localized in the vicinity of the impurity can be tuned into resonance with chain-extended states by adjusting $w(N)\ensuremath{\propto}{N}^{1/2}$; the resulting avoided crossing yields resonant eigenstates with large amplitudes on the chain end points and approximately equidistant eigenvalues. The state transfer time scales as $t\ensuremath{\propto}{N}^{3/2}$, and its transfer probability PPP approaches unity in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$; the error scales as $1\ensuremath{-}P\ensuremath{\propto}{N}^{\ensuremath{-}1}$. Thus, with the addition of two impurities, asymptotically perfect efficient state transfer with a uniform chain is possible even in the absence of external control.
Efficient quantum-state transfer is achieved in a uniformly coupled spin-1/21/21/2 chain, with open boundaries, by application of local magnetic fields on the second and last-but-one spins, respectively. These effective barriers induce the appearance of two eigenstates, bilocalized at the edges of the chain, which allow a high-quality transfer also at relatively long distances. The same mechanism may be used to send an entire e-bit (e.g., an entangled qubit pair) from one to the other end of the chain.
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.
We investigate the fidelity of the quantum state transfer (QST) of two qubits by means of an arbitrary spin-12\frac{1}{2}21​ network, on a lattice of any dimensionality. Under the assumptions that the network Hamiltonian preserves the magnetization and that a fully polarized initial state is taken for the lattice, we obtain a general formula for the average fidelity of the two qubits QST, linking it to the one- and two-particle transfer amplitudes of the spin excitations among the sites of the lattice. We then apply this formalism to a 1D spin chain with XXXXXX-Heisenberg type nearest-neighbour interactions adopting a protocol that is a generalization of the single qubit one proposed in Paganelli et al. [Phys. Rev. A 87, 062309 (2013)]. We find that a high-quality two qubit QST can be achieved provided one can control the local fields at sites near the sender and receiver. Under such conditions, we obtain an almost perfect transfer in a time that scales either linearly or, depending on the spin number, quadratically with the length of the chain.