We have now seen how processes described by string diagrams already exhibit some quantum-like features. It is natural to ask how much extra work is needed to go from string diagrams to Hilbert spaces and linear maps, the mathematical tools von Neumann used to formulate quantum theory in the late 1920s. The answer is: not that much.

We start by considering what it takes for two processes to be equal. In many process theories, it suffices for them to agree on a relatively small number of states. This feature leads very naturally to the notion of basis, and we can use adjoints to identify a particularly handy type of basis, called an orthonormal basis (ONB). When all types admit a basis, any process can be completely described by a collection of numbers called its matrix.

Now, such a matrix identifies a particular process uniquely, but for any matrix to represent a process we need to add a bit more structure. Therefore, we allow processes with the same input/output wires to be combined into one, or summed together. If a process theory admits string diagrams, has ONBs for every type, and has sums of processes, we can describe sums, sequential composition, parallel composition, transpose, conjugate, and adjoint all in terms of operations on matrices. We call this the matrix calculus of a process theory.

Thus, by adding ONBs and sums, we have very nearly recovered the full power of linear algebra, but with the added generality that the numbers λ are still very unrestricted (in particular, they need not be the elements of some field like the real or complex numbers). In fact, a matrix calculus for relations makes perfect sense, where the numbers are booleans.

The final step towards Hilbert spaces and linear maps consists of requiring the numbers of the process theory to be the complex numbers.