1 Introduction

This quantum information package for the Maxima computer algebra system allows the manipulation of instances of objects-- operators, vectors, tensors, etc. that appear in the theory of quantum information and of quantum entanglement. More precisely these objects are typically represented in this package as row and column vectors and matrices, whose entries may be explicit numbers (of various classes) or algebraic expressions. This software occupies a niche distinct from high performance numerical linear algebra software as well as software such as the Maxima tensor packages that manipulate abstract mathematical objects. This document describes the functions and data in the package and how to use them with Maxima , assuming that you do not know much about Maxima , but do know quantum information theory. However, most of the examples are also found in introductory texts on quantum information. The package is intended for research and teaching in the theory of entanglement and quantum information and related fields.

Examples of the facilities of the package are

• Methods for constructing pure and mixed states and operators.
• Methods for executing standard operations found in computational linear algebra as well as the tensor product, partial trace, etc.
• Functions to compute commonly appearing quantities such as entropy and purity.

This document begins with a very brief introduction to Maxima emphasizing features that are important for qinf. The remainder of the manual is a series of small sections introducing functions with examples. The examples mostly consist of testing equations. First identities and textbook exercises are presented, in part because they are the natural calculations to include in the test suite from which they are taken. Then more complicated calculations are tested, as that is the initial application of the author-- to check manual calculations and results found in other documents. In this restricted sense, the package can give results on simple abstract statements: If 1) generic instances of objects are generated, and 2) a representation-invariant statement is formulated, and 3) the subexpressions are successfully coerced into some canonical form, then defects in the statement can sometimes be detected if the statement is not true.

Some suggestions and things to be aware of in the following sections.

• You probably need to read at least a ten minute Maxima tutorial before or in conjunction with reading this document. There are several listed at the Maxima website, and others that a search engine can find. If you are too impatient there is a very brief introduction to Maxima below. The best source for most questions is the Maxima manual.
• There are several user interfaces to Maxima . All the examples here are generated using the imaxima package for the emacs editor/environment, but the results are similar to other graphical frontends to Maxima . Other popular graphical frontends are wxmaxima, and texmacs.
• Most functions currently work only with qubits, others for variable number of states.

1.1 Acknowledgments

Some of the ideas used in qinf are inspired by the package qdensity written for a proprietary symbolic algebra system. None of the code in qdensity has been borrowed for this project, however. Advice on Maxima and lisp programming was provided by, among others, Robert Dodier, Stavros Macrakis, and Barton Willis.