A quantum protocol to win the graph colouring game on all Hadamard graphs

This paper deals with graph colouring games, an example of pseudo-telepathy, in which two provers can convince a verifier that a graph $G$ is $c$-colourable where $c$ is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph $G_N$ with $N=c=16$. Their protocol applies only to Hadamard graphs where $N$ is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of $G_{12}$ having 1609 vertices, with $c=12$. Combined with a result of Frankl and Rodl, our result shows that all sufficiently large Hadamard graphs yield pseudo-telepathy games.Comment: 5pag

Configuration determination for chiral and polar crystals by anisotropic NMR shift

We propose a method to perform a configuration determination for chiral and/or polar crystals by utilizing anisotropy of NMR shift. The chirality (handedness) or polarity of a crystal, that is characterized by its sign, can be extracted from the asymmetric magnetic-field angle dependence of NMR shift in the appropriate plane, as the configuration is reflected in the off-diagonal components of the shift tensor. This method is applicable to the triclinic, monoclinic, and trigonal crystal structures among 230 crystallographic space groups, and the appropriate planes to extract the asymmetric field-angle dependence are tabulated for all relevant space groups. We discuss how to determine the appropriate plane, and to identify each contribution of twin domains, and so on, by using the specific examples of Te, IrSn$_{4}$ and RhSn$_{4}$, and the spontaneous symmetry-breaking phases of URhSn. We also argue that an absolute-configuration determination is also possible in accordance with the relation between chirality or polarity of crystal and the sign of the relevant off-diagonal component of the shift tensor, provided by theoretical evaluation of internal magnetic field from surrounding ions, and/or by experimental input on the shape of the asymmetric etch pit on the cleaved face of crystal.Comment: 12 pages, 7 figures, 5 table