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arXiv:2407.16460 · Lobachevskii Journal of Mathematics

Some Aspects of Remote State Restoring in State Transfer Governed by XXZ-Hamiltonian

G. A. Bochkin, Serge I. Doronin, Edward B. Fel’dman, E. I. Kuznetsova, I. D. Lazarev, Alexander Nikolaevich Pechen, Alexander Ivanovich Zenchuk

Published Mar 1, 2024• 3 citations
arXivHTMLPDF

Abstract

Abstract We consider the remote state restoring and perfect transfer of the zero-order coherence matrix (PTZ) in a spin system governed by the XXZ-Hamiltonian conserving the excitation number. The restoring tool is represented by several nonzero Larmor frequencies in the Hamiltonian. To simplify the analysis we use two approximating models including either step-wise or pulse-type time-dependence of the Larmor frequencies. Restoring in spin chains with up to 20 nodes is studied. Studying PTZ, we consider the zigzag and rectangular configurations and optimize the transfer of the 0-order coherence matrix using geometrical parameters of the communication line as well as the special unitary transformation of the extended receiver. Overall observation is that XXZ-chains require longer time for state transfer than XX-chains, which is confirmed by the analytical study of the evolution under the nearest-neighbor approximation. We demonstrate the exponential increase of the state-transfer time with the spin chain length.

DOI
10.1134/s1995080224600596

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Abstract We consider the remote state restoring and perfect transfer of the zero-order coherence matrix (PTZ) in a spin system governed by the XXZ-Hamiltonian conserving the excitation number. The restoring tool is represented by several nonzero Larmor frequencies in the Hamiltonian. To simplify the analysis we use two approximating models including either step-wise or pulse-type time-dependence of the Larmor frequencies. Restoring in spin chains with up to 20 nodes is studied. Studying PTZ, we consider the zigzag and rectangular configurations and optimize the transfer of the 0-order coherence matrix using geometrical parameters of the communication line as well as the special unitary transformation of the extended receiver. Overall observation is that XXZ-chains require longer time for state transfer than XX-chains, which is confirmed by the analytical study of the evolution under the nearest-neighbor approximation. We demonstrate the exponential increase of the state-transfer time with the spin chain length.
Abstract Optimal state transport across spin chains, which are proposed as quantum wires for information transfer in solid state quantum architectures, is an important topic for quantum technologies. In this work, we study the remote restoring of a quantum state transferred along a spin chain. The structural state-restoring technique provides proportionality between the appropriate elements of the density matrices of the initial sender state and receiver state at some time instant. We develop a remote state-restoring protocol which uses an inhomogeneous magnetic field with step-wise time-dependent Larmor frequencies as the state-control tool. For simulating the multiparametric Hamiltonian we use two approximating models. First model is based on the Trotter-Suzuki method, while the second model is based on using short pulses of high intensity. In both cases we estimate the accuracy of the approximation and find the optimal restoring parameters (Larmor frequencies) of the protocol which maximize the coefficients in the proportionality for spin chains of various lengths.
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.
Abstract We propose a protocol for restoring states transferred along a spin-1/2 chain utilizing a controlled interaction of the spin-chain with the environment. Such interaction exploits an incoherent quantum control and is described by the master equation with Lindblad operators having time-dependent pulse-sequence form. The amplitudes of the pulses are properly adjusted to perform the state restoring. We apply this protocol to restore the 1-order coherence matrix of the (0,1)-excitation of a 2-qubit state transferred along the spin-1/2 chain governed by the XXZ-Hamiltonian preserving the excitation number in the system.
We study the long distance [Formula: see text]-excitation state restoring in the linear open chain governed by the XX-Hamiltonian. We show that restoring the 1-order coherence matrix results in restoring the 1-excitation block of the 0-order coherence matrix, so that only one 0-excitation element of the density matrix remains unrestored. Such restoring also scales the concurrence between any two qubits of the transferred state, the scaling factor is defined by the Hamiltonian and doesn’t depend on the initial sender’s state. Sender–Receiver entanglement is also studied via the PPT criterion.
We propose a protocol restoring the state transferred along the spin chain governed by the XY-Hamiltonian. Such dynamics does not preserve the excitation number and leads to mixing multiple-quantum coherence matrices of orders having the same parities. Depending on the initial state, this results in generating either all possible or only even-order multiple-quantum coherence matrices. Restoring is established via the special unitary transformation applied to the receiver side of the chain (extended receiver). An example of restoring [Formula: see text]- and [Formula: see text]-order coherence matrices in two-qubit state transfer is considered. Entanglement transfer in such process is also studied and possibility of its amplification is demonstrated.
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.