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arXiv:0901.3424 · Physical Review A

Relationship between probabilities of the state transfers and entanglements in spin systems with simple geometrical configurations

Serge I. Doronin, Edward B. Fel’dman, Alexander Ivanovich Zenchuk

Published Apr 8, 2009• 16 citations
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Abstract

In this paper we derive analytical relations between probabilities of the excited state transfers and entanglements calculated by both the Wootters and positive partial transpose criteria for the arbitrary spin system with single excited spin in the external magnetic field and Hamiltonian commuting with ${I}_{z}$. We apply these relations to study the arbitrary state transfers and entanglements in the simple systems of nuclear spins having two- and three-dimensional geometrical configurations with $XXZ$ Hamiltonian. It is shown that high-probability state transfers (HPSTs) are possible among all four nodes placed in the corners of the rectangle with the proper ratio of sides as well as among all eight nodes placed in the corners of the parallelepiped with the proper ratio of sides. Entanglements responsible for these HPSTs have been identified.

DOI
10.1103/physreva.79.042310

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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.
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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.
Entanglement is studied in an open alternating chain of nuclear spins s = 1/2 with spin-spin couplings in an external magnetic field under the thermodynamic equilibrium conditions. A reduced density matrix has been calculated for an arbitrarily chosen spin pair. The entanglement of the spin pair has been estimated according to the Wootters criterion. The temperature at which an entangled state appears in a chosen pair has been determined. It has been shown that the numerical characteristics of the entanglement are oscillating functions of the position of the spin pair in the chain.
The geometrical significance of winding number is used to study the topological characteristics of a single-excitation Heisenberg XY spin model represented by the Su-Schrieffer-Heeger (SSH) model. Due to the presence of topological edge states, the quantum state may be conveyed from the first site to the last site with a high degree of fidelity. Also examined is a Heisenberg XY model on the kagome lattice. In the topological phase of the kagome lattice, the three corner states are not affected by the Dzyaloshinsky-Moriya (DM) interactions. For the kagome lattice with and without the DM interactions, the state of a spin on one corner site can be transferred to the other two corner sites in various ways during the topological phase. The quantum state transfer (QST) for the XY model on the kagome lattice without the DM interactions is symmetrical and can only produce a three-body entangled state. In contrast, the QST for the XY model on the kagome lattice with the DM interactions is in an asymmetrical form, similar to one-way propagation, and it can produce both product states and three-body entangled states.
The probability amplitudes of the processes related to the transfer of the excited state from one qubit to another are calculated using an indirect interaction in an open waveguide. The system consists of two qubits located at an arbitrary distance from each other. The non-Hermitian effective Hamiltonian approach used herein makes it possible to bypass using the Markovchain approximation method. Analytic expressions describing the probability of transfer of the excited state from one qubit to another under different initial states of the system were obtained.