Permutations over cyclic groups

Generalizing a result in the theory of finite fields we prove that, apart from a couple of exceptions that can be classified, for any elements $a_1,...,a_m$ of the cyclic group of order $m$, there is a permutation $\pi$ such that $1a_{\pi(1)}+...+ma_{\pi(m)}=0$

Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4

We work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic 0 eigenform. The weak eigenform is closely related to Ramanujan's tau function whereas the characteristic 0 eigenform is attached to an elliptic curve defined over ${\mathbb Q}$. We produce the lift by showing that the coefficients of the initial, weak eigenform (almost all) occur as traces of Frobenii in the Galois representation on the 4-torsion of the elliptic curve. The example is remarkable as the initial form is known not to be liftable to any characteristic 0 eigenform of level 1. We use this example as illustrating certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions

Long-distance quantum communication over noisy networks without long-time quantum memory

The problem of sharing entanglement over large distances is crucial for implementations of quantum cryptography. A possible scheme for long-distance entanglement sharing and quantum communication exploits networks whose nodes share Einstein-Podolsky-Rosen (EPR) pairs. In Perseguers et al. [Phys. Rev. A 78, 062324 (2008)] the authors put forward an important isomorphism between storing quantum information in a dimension $D$ and transmission of quantum information in a $D+1$-dimensional network. We show that it is possible to obtain long-distance entanglement in a noisy two-dimensional (2D) network, even when taking into account that encoding and decoding of a state is exposed to an error. For 3D networks we propose a simple encoding and decoding scheme based solely on syndrome measurements on 2D Kitaev topological quantum memory. Our procedure constitutes an alternative scheme of state injection that can be used for universal quantum computation on 2D Kitaev code. It is shown that the encoding scheme is equivalent to teleporting the state, from a specific node into a whole two-dimensional network, through some virtual EPR pair existing within the rest of network qubits. We present an analytic lower bound on fidelity of the encoding and decoding procedure, using as our main tool a modified metric on space-time lattice, deviating from a taxicab metric at the first and the last time slices.Comment: 15 pages, 10 figures; title modified; appendix included in main text; section IV extended; minor mistakes remove

A coprimality condition on consecutive values of polynomials

Let $f\in\mathbb{Z}[X]$ be quadratic or cubic polynomial. We prove that there exists an integer $G_f\geq 2$ such that for every integer $k\geq G_f$ one can find infinitely many integers $n\geq 0$ with the property that none of $f(n+1),f(n+2),\dots,f(n+k)$ is coprime to all the others. This extends previous results on linear polynomials and, in particular, on consecutive integers