In Chapters 2 and 3 we assumed a model according to which an experiment is split into two parts – preparation and measurement. This led us to the associated concepts of states and observables, respectively. On the basis of this picture we can think about two types of apparatus, which can be characterized abstractly as follows.

• Preparators are devices producing quantum states. No input is required, but a quantum output is produced.

• Measurements are input–output devices that accept a quantum system as their input and produce a classical output in the form of measurement outcome distributions.

This setting is summarized in Figure 2.2. Furthermore, in subsection 3.5.2 we discussed coarse-graining relations; coarse-graining can be depicted as a box with a classical input and a classical output. This line of thought indicates that at least one type of input–output box is so far missing from our discussion – a device taking a quantum input and producing a quantum output. Hence, we say that

•quantum channels are input–output devices transforming quantum states into quantum states.

Each quantum channel can be placed between arbitrary preparation and measurement devices (Figure 4.1). Channels, and a slightly more general concept, that of operations, are the topics of this chapter.

Transforming quantum systems

In this section we discuss the mathematical definition of operations and channels. The notion of an operation was introduced by Haag and Kastler [70] within the framework of algebraic quantum field theory. Later Kraus [88], Lindblad [93] and Davies [51] discussed the requirement for operations to be completely positive.