Throughout this text we assume that the reader is familiar with elementary quantum mechanics and the properties of complex vector spaces, and in this appendix we provide a brief reminder of these topics. In particular, we introduce Dirac's notation for describing quantum mechanical systems. Many areas of quantum mechanics studied in undergraduate degrees can be described without using Dirac notation, and its importance is unclear. In other areas, however, the advantages of Dirac notation are huge, and it is essentially the only notation in use. This is particularly true of quantum information theory.

Dirac's notation is closely related to that used to describe abstract vector spaces known as Hilbert spaces, and many formal arguments about the properties of quantum systems are in fact arguments about the properties of Hilbert spaces. Here we aim to steer a careful course between the twin perils of excessive mathematical sophistication and of taking too much on trust. We will not prove some elementary results whose proof can be found elsewhere, but will concentrate on how these results can be used.

Hilbert space

A Hilbert space is an abstract vector space. As such, it has many properties in common with the use of ordinary three-dimensional vectors, but it also differs in several important ways. Firstly, the vector space is not three-dimensional, but can have any number of dimensions. (The description below largely assumes that the number of dimensions is finite, but it is also possible to extend these results to infinite-dimensional spaces.) Secondly, when the vectors are multiplied by scalar numbers these numbers can be complex.