Generalized Multiplicative Domains and Quantum Error Correction

Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi's characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new representation-theoretic description of quantum error-correcting codes that extends previous results for unitarily-correctable codes, noiseless subsystems and decoherence-free subspaces.Comment: 14 page

Homotopy hyperbolic 3-manifolds are hyperbolic

This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold

The electrical response matrix of a regular 2n-gon

Consider a unit-resistive plate in the shape of a regular polygon with 2n sides, in which even-numbered sides are wired to electrodes and odd-numbered sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows one to compute the currents flowing through the electrodes when they are held at specified voltages. We show that the entries of the response matrix of the regular 2n-gon are given by the differences of cotangents of evenly spaced angles, and we describe some connections with the limiting distributions of certain random spanning forests.Comment: 10 pages, 4 figures; v2 adds more background informatio

Intercalated Rare-Earth Metals under Graphene on SiC

Intercalation of rare earth metals ($RE$ = Eu, Dy, and Gd) is achieved by depositing the $RE$ metal on graphene that is grown on silicon-carbide (SiC) and by subsequent annealing at high temperatures to promote intercalation. STM images of the films reveal that the graphene layer is defect free and that each of the intercalated metals has a distinct nucleation pattern. Intercalated Eu forms nano-clusters that are situated on the vertices of a Moir{\`e} pattern, while Dy and Gd form randomly distributed nano-clusters. X-ray magnetic circular dichroism (XMCD) measurements of intercalated films reveal the magnetic properties of these $RE$'s nano-clusters. Furthermore, field dependence and temperature dependence of the magnetic moments extracted from the XMCD show paramagnetic-like behaviors with moments that are generally smaller than those predicted by the Brillouin function. XMCD measurements of $RE$-oxides compared with those of the intercalated $RE$'s under graphene after exposure to air for months indicate that the graphene membranes protect these intercalants against oxidation.Comment: 9 pages, 7 figure