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arXiv:0811.0339 · Journal of Physics A Mathematical and Theoretical

Designing lattice structures with maximal nearest-neighbor entanglement

Jorge Carlos Navarro-Muñoz, Román López-Sandoval, Martı́n E. Garcia

Published Jul 15, 2009• 1 citation

Abstract

In this paper, we study the numerical optimization of nearest-neighbor concurrence of bipartite one- and two-dimensional lattices, as well as non-bipartite two-dimensional lattices. These systems are described in the framework of a tight-binding Hamiltonian while the optimization of concurrence was performed using genetic algorithms. Our results show that the concurrence of the optimized lattice structures is considerably higher than that of non-optimized systems. In the case of one-dimensional chains, the concurrence increases dramatically when the system begins to dimerize, i.e., it undergoes a structural phase transition (Peierls distortion). This result is consistent with the idea that entanglement is maximal or shows a singularity near quantum phase transitions. Moreover, the optimization of concurrence in two-dimensional bipartite and non-bipartite lattices is achieved when the structures break into smaller subsystems, which are arranged in geometrically distinguishable configurations.

DOI: 10.1088/1751-8113/42/31/315302

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In this paper, we study the numerical optimization of nearest-neighbor concurrence of bipartite one- and two-dimensional lattices, as well as non-bipartite two-dimensional lattices. These systems are described in the framework of a tight-binding Hamiltonian while the optimization of concurrence was performed using genetic algorithms. Our results show that the concurrence of the optimized lattice structures is considerably higher than that of non-optimized systems. In the case of one-dimensional chains, the concurrence increases dramatically when the system begins to dimerize, i.e., it undergoes a structural phase transition (Peierls distortion). This result is consistent with the idea that entanglement is maximal or shows a singularity near quantum phase transitions. Moreover, the optimization of concurrence in two-dimensional bipartite and non-bipartite lattices is achieved when the structures break into smaller subsystems, which are arranged in geometrically distinguishable configurations.

We consider translationally invariant states of an infinite one-dimensional chain of qubits or spin-

\frac{1}{2}

particles. We maximize the entanglement shared by nearest neighbors via a variational approach based on finitely correlated states. We find an upper bound of nearest-neighbor concurrence equal to

\mathcal{C}=0.434\phantom{\rule{0.2em}{0ex}}095

which is

0.09%

away from the bound

{\mathcal{C}}_{\mathrm{W}}=0.434\phantom{\rule{0.2em}{0ex}}467

obtained by a completely different procedure. The obtained state-maximizing nearest-neighbor entanglement seems to approximate the maximally entangled mixed states. Further we investigate in detail several other properties of the so-obtained optimal state.

The pairwise quantum entanglement between sites in disordered electronic one-dimensional systems (rings) is studied. We focus on the effect of diagonal and off-diagonal disorder on the ground-state concurrence

{C}_{ij}

between neighboring and not neighboring sites

i,j

as a function of electron band filling. In the case of diagonal disorder, increasing the degree of disorder leads to a decrease of the concurrence with respect to the ordered case. However, off-diagonal disorder causes a surprisingly strong enhancement of nearest-neighbor entanglement. This remarkable effect occurs near half filling, where the concurrence becomes up to 15% larger than in the periodic system.

The features of the concurrences between the nearest-neighbors and that of the next-nearest-neighbors for the one-dimensional Heisenberg model with next-nearest-neighbor coupling

J

are studied as functions of temperature and

J

. The two concurrences exhibit a different dependence on

J

at the ground state, which could be interpreted from the point of view of the correlation functions. The threshold temperature at which the concurrence is zero and the temperature effect on the two concurrences for systems up to 12 sites are studied numerically.

copies of a partially entangled state, a single state with higher entanglement can be obtained. Obtaining a maximally entangled state is possible for N=1. However, the associated success probability can be extremely low when increasing the system's dimensionality. In this work, we study two methods to achieve a probabilistic entanglement concentration for bipartite quantum systems with a large dimensionality for N=1, regarding a reasonably good probability of success at the expense of having a non-maximal entanglement. Firstly, we define an efficiency function Q considering a tradeoff between the amount of entanglement (quantified by the I-Concurrence) of the final state after the concentration procedure and its success probability, which leads to solving a quadratic optimization problem. We found an analytical solution, ensuring that an optimal scheme for entanglement concentration can always be found in terms of Q. Finally, a second method was explored, which is based on fixing the success probability and searching for the maximum amount of entanglement attainable. Both ways resemble the Procrustean method applied to a subset of the most significant Schmidt coefficients but obtaining non-maximally entangled states.

Entanglement sharing among pairs of spins in Heisenberg antiferromagnets is investigated using the concurrence measure. For a nondegenerate

S=0

ground state, a simple formula relates the concurrence to the diagonal correlation function. The concurrence length is seen to be extremely short. Various finite clusters are studied numerically, to see the trend in two dimensions. It is argued that the concurrences vanish between pairs of spins which are not nearest neighbors for the linear-chain and square-lattice antiferromagnet ground states. For the triangular lattice and Kagome' lattice, nearest-neighbor concurrences also vanish. The concurrences in the maximal-spin states are explicitly calculated, where the concurrence averaged over all pairs is larger than the

S=0

states.