Central symmetry in two-dimensional lattices and quantum information transmission

Abstract

We present a method to numerically calculate the dynamics driven by arbitrary Hamiltonians, in particular those with symmetry. The method enables us to study quantum state and entanglement transmission through a uniformly coupled two-dimensional square lattice with XK-type interaction. Significantly, we find that the central symmetry between sending and receiving nodes plays a crucial role in attaining maximal fidelity and entanglement transmission. In other words, the quality of a two-dimensional transmission is determined by the geometry of the nodes involved.Transmission of quantum states from one location to another is often required in quantum information processing. One of the natural mediators for such a task is a spin chain [1-10]. The motivation for using an unmodulated Heisenberg spin chain dates back to Bose's work [1]. Later literature shows that a quantum state can be transferred perfectly by properly preengineering the couplings between nearestneighbor sites [2].Antiferromagnetic XY models for quantum state transmission (QST) have been paid special attention (see, e.g., [3,9]) because of their simplicity and symmetry. High-fidelity QST can be obtained by encoding multiple-spin quantum states [5] in these models. A physical explanation for obtaining such high fidelity is given in recent literature [10][11][12]. These publications also present a method for the numerical determination of good initial states which result in high-fidelity state transmission or even perfect state transfer [2].However, most previous QST literature concems onedimensional spin chains, although QSTs on two-and threedimensional lattices have also received attention sporadically [13][14][15][16]. In Ref.[13] analytic models that can perfectly transfer arbitrary quantum states in two-and three-dimensional interacting bosonic and fermionic networks are found by properly engineering the couplings. In Ref.[14] QSTs through a two-dimensional regular spin lattice with nonuniform coupling are analyzed and high fidelity QST is obtained. QSTs in a twodimensional hexagonal lattice and a three-dimensional lattice are investigated in Ref. [15]. Quantum state and entanglement transmission through a two-dimensional spin network with periodic boundary conditions may have an analytical solution and has been investigated recently [16]. Entanglement properties in a two-dimensional triangular lattice with impurities are investigated in Refs.[17] and [18], where Ising and XY-type interactions are considered, respectively. Experimentally it is hard to realize the required nonuniform couplings between sites, whereas uniform couplings between spin sites are much more feasible with state-of-the-art technologies.Most spin models do not have analytical solutions, in particular higher-dimensional ones, for instance, the twodimensional nearest-neighbor XY model with uniform couplings. Unlike analytical approaches, this paper presents a method to exactly and numerically calculate single-excitation dynamics of a two-dimensional nearest-neighbor XY model wit...