Most quantum protocols rely on the interplay between quantum systems and classical data. For instance, measurements extract classical data from a quantum system, whereas controlled operations use classical data to affect a quantum system. Moreover, given that truth is in the eye of the beholder, we want to understand quantum theory relative to our perception of reality, which is classical, and hence want to understand how the two relate. Somewhat surprisingly, it turns out to be much easier to represent the classical world relative to quantum processes, than the other way around.

One way to get a handle on this interaction is to express as much of it as possible in a purely diagrammatic form. Previously, we have drawn diagrams of quantum processes, then used some sort of ‘external’ means of describing the classical data flow, i.e. with indices and brackets, which can't really be plugged together like pieces of a diagram. Even worse, in most standard textbooks, classical data is not even part of the actual formalism, but is described in words.

Rather than describing this interplay of quantum systems and classical data using lots of ‘blah blah blah’, or a cross-breed between diagrams and symbols, can we instead just give a diagram of all of the devices involved and how they are wired together? For example, suppose we have a device ‘Bell’ that prepares Bell states, another device ‘Bell-M’ that performs Bell measurements, and a third device ‘Bell-C’ that does Bell corrections, depicted very realistically as follows:

Now, suppose we want to describe to a technician how to wire these devices together to do teleportation:

We could describe this using a specification language, that is, a diagrammatic language where the boxes correspond to devices and the wires correspond to literally ‘wiring up’ the devices:

where we now distinguish quantum wires and classical wires.