No abstract

We analyze the decoherence in quantum walks in two-dimensional lattices generated by brokenlink-type noise. In this type of decoherence, the links of the lattice are randomly broken with some given constant probability. We obtain the evolution equation for a quantum walker moving on 2-D lattices subject to this noise, and we point out how to generalize for lattices in more dimensions. In the non-symmetric case, when the probability to break links in one direction is different from the probability in the perpendicular direction, we have obtained a non-trivial result. If one fixes the link-breaking probability in one direction, and gradually increases the probability in the other direction from 0 to 1, the decoherence initially increases until it reaches a maximum value, and then it decreases. This means that, in some cases, one can increase the noise level and still obtain more coherence. Physically, this can be explained as a transition from a decoherent 2-D walk to a coherent 1-D walk.

The Invar package is introduced, a fast manipulator of generic scalar polynomial expressions formed from the Riemann tensor of a fourdimensional metric-compatible connection. The package can maximally simplify any polynomial containing tensor products of up to seven Riemann tensors within seconds. It has been implemented both in Mathematica and Maple algebraic systems. Program summaryProgram title: Invar Tensor Package Catalogue identifier: ADZK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZK_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 136 240 No. of bytes in distributed program, including test data, etc.: 2 711 923 Distribution format: tar.gz Programming language: Mathematica and Maple Computer: Any computer running Mathematica versions 5.0 to 5.2 or Maple versions 9 and 10 Operating system: Linux, Unix, Windows XP RAM: 30 Mb Word size: 64 or 32 bits Classification: 5External routines: The Mathematica version requires the xTensor and xPerm packages. These are freely available at http://metric.iem.csic.es/MartinGarcia/xAct Nature of problem: Manipulation and simplification of tensor expressions. Special attention on simplifying scalar polynomial expressions formed from the Riemann tensor on a four-dimensional metric-compatible manifold. Solution method: Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor. Restrictions: The present versions do not fully address the problem of reducing differential invariants or monomials of the Riemann tensor with free indices. Running time: Less than a second to fully reduce a monomial of the Riemann tensor of degree 7 in terms of independent invariants.

The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6x10^23 objects with up to 12 derivatives of the metric. This covers cases ranging from products of up to 6 undifferentiated Riemann tensors to cases with up to 10 covariant derivatives of a single Riemann. We extend our computer algebra system Invar to produce within seconds a canonical form for any of those objects in terms of a basis. The process is as follows: (1) an invariant is converted in real time into a canonical form with respect to the permutation symmetries of the Riemann tensor; (2) Invar reads a database of more than 6x10^5 relations and applies those coming from the cyclic symmetry of the Riemann tensor; (3) then applies the relations coming from the Bianchi identity, (4) the relations coming from commutations of covariant derivatives, (5) the dimensionally-dependent identities for dimension 4, and finally (6) simplifies invariants that can be expressed as product of dual invariants. Invar runs on top of the tensor computer algebra systems xTensor (for Mathematica) and Canon (for Maple).Comment: 12 pages, 1 figure, 3 tables. Package can be downloaded from http://metric.iem.csic.es/Martin-Garcia/xAct/Invar/ (Mathematica version) or http://www.lncc.br/~portugal/Invar.html (Maple version