SNEG - Mathematica package for symbolic calculations with second-quantization-operator expressions

In many-particle problems involving interacting fermions or bosons, the most natural language for expressing the Hamiltonian, the observables, and the basis states is the language of the second-quantization operators. It thus appears advantageous to write numerical computer codes which allow the user to define the problem and the quantities of interest directly in terms of operator strings, rather than in some low-level programming language. Here I describe a Mathematica package which provides a flexible framework for performing the required translations between several different representations of operator expressions: condensed notation using pure ASCII character strings, traditional notation ("pretty printing"), internal Mathematica representation using nested lists (used for automatic symbolic manipulations), and various higher-level ("macro") expressions. The package consists of a collection of transformation rules that define the algebra of operators and a comprehensive library of utility functions. While the emphasis is given on the problems from solid-state and atomic physics, the package can be easily adapted to any given problem involving non-commuting operators. It can be used for educational and demonstration purposes, but also for direct calculations of problems of moderate size.Comment: 9 pages, 1 figur

Codes, orderings, and partial words

Codes play an important role in the study of the combinatorics of words. In this paper, we introduce pcodes that play a role in the study of combinatorics ofpartial words. Partial words are strings over a finite alphabet that may contain a number of “do not know” symbols. Pcodes are defined in terms of the compatibility relation that considers two strings over the same alphabet that are equal except for a number of insertions and/or deletions of symbols. We describe various ways of defining and analyzing pcodes. In particular, many pcodes can be obtained as antichains with respect to certain partial orderings. Using a technique related to dominoes, we show that the pcode property is decidable

Embedding a θ\theta-invariant code into a complete one

Let A be a finite or countable alphabet and let $\theta$ be a literal (anti-)automorphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short) that is, languages L such that $\theta$ (L) is a subset of L.We establish an extension of the famous defect theorem. With regards to the so-called notion of completeness, we provide a series of examples of finite complete $\theta$-invariant codes. Moreover, we establish a formula which allows to embed any non-complete $\theta$-invariant code into a complete one. As a consequence, in the family of the so-called thin $\theta$--invariant codes, maximality and completeness are two equivalent notions.Comment: arXiv admin note: text overlap with arXiv:1705.0556

Algebraic Approach to Physical-Layer Network Coding

The problem of designing physical-layer network coding (PNC) schemes via nested lattices is considered. Building on the compute-and-forward (C&F) relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, an algebraic approach is taken to show its potential in practical, non-asymptotic, settings. A general framework is developed for studying nested-lattice-based PNC schemes---called lattice network coding (LNC) schemes for short---by making a direct connection between C&F and module theory. In particular, a generic LNC scheme is presented that makes no assumptions on the underlying nested lattice code. C&F is re-interpreted in this framework, and several generalized constructions of LNC schemes are given. The generic LNC scheme naturally leads to a linear network coding channel over modules, based on which non-coherent network coding can be achieved. Next, performance/complexity tradeoffs of LNC schemes are studied, with a particular focus on hypercube-shaped LNC schemes. The error probability of this class of LNC schemes is largely determined by the minimum inter-coset distances of the underlying nested lattice code. Several illustrative hypercube-shaped LNC schemes are designed based on Construction A and D, showing that nominal coding gains of 3 to 7.5 dB can be obtained with reasonable decoding complexity. Finally, the possibility of decoding multiple linear combinations is considered and related to the shortest independent vectors problem. A notion of dominant solutions is developed together with a suitable lattice-reduction-based algorithm.Comment: Submitted to IEEE Transactions on Information Theory, July 21, 2011. Revised version submitted Sept. 17, 2012. Final version submitted July 3, 201