Better Lattice Quantizers Constructed from Complex Integers

Real-valued lattices and complex-valued lattices are mutually convertible, thus we can take advantages of algebraic integers to defined good lattice quantizers in the real-valued domain. In this paper, we adopt complex integers to define generalized checkerboard lattices, especially $\mathcal{E}_{m}$ and $\mathcal{E}_{m}^+$ defined by Eisenstein integers. Using $\mathcal{E}_{m}^+$, we report the best lattice quantizers in dimensions $14$, $18$, $20$, and $22$. Their product lattices with integers $\mathbb{Z}$ also yield better quantizers in dimensions $15$, $19$, $21$, and $23$. The Conway-Sloane type fast decoding algorithms for $\mathcal{E}_{m}$ and $\mathcal{E}_{m}^+$ are given.Comment: 7 page

Quantum Hall Effects in a Non-Abelian Honeycomb Lattice

We study the tunable quantum Hall effects in a non-Abelian honeycomb optical lattice which is a many-Dirac-points system. We find that the quantum Hall effects present different features as change as relative strengths of several perturbations. Namely, a gauge-field-dressed next-nearest-neighbor hopping can induce the quantum spin Hall effect and a Zeeman field can induce a so-called quantum anomalous valley Hall effect which includes two copies of quantum Hall states with opposite Chern numbers and counter-propagating edge states. Our study extends the borders of the field of quantum Hall effects in honeycomb optical lattice when the internal valley degrees of freedom enlarge.Comment: 7 pages, 6 figure

Improved lower bounds on genuine-multipartite-entanglement concurrence

Genuine-multipartite-entanglement (GME) concurrence is a measure of genuine multipartite entanglement that generalizes the well-known notion of concurrence. We define an observable for GME concurrence. The observable permits us to avoid full state tomography and leads to different analytic lower bounds. By means of explicit examples we show that entanglement criteria based on the bounds have a better performance with respect to the known methods.Comment: 17 pages, 1 EPS figure; v3 is in one column to improve readability of equation