## 0.6 Operators

A **linear map** between two vector spaces **endomorphisms**, that is, maps from **operators**.
The symbol *does* matter: in general, **commute**.
The inverse of *one* of these two conditions, since any one of the two implies the other, whereas, on an infinite-dimensional space, *both* must be checked.
Finally, given a particular basis, an operator

The **adjoint**, or **Hermitian conjugate**, of an linear map

An operator

**normal**ifAA^\dagger = A^\dagger A **unitary**ifA^\dagger=A^{-1} **Hermitian**(or**self-adjoint**) ifA^\dagger = A .

In particular then, being unitary implies being normal, since if *operators*, i.e. they are represented by a *square* matrix.

Any *physically admissible* evolution of an isolated quantum system is represented by a unitary operator.^{14}
Note that unitary operators preserve the inner product: given a unitary operator *unitary operations are the isometries of the Euclidean norm*.

This whole package of stuff and properties and structure (i.e. finite dimensional Hilbert spaces with linear maps and the dagger) bundles up into an abstract framework called a **dagger compact category**.
We will not delve into the vast world of category theory in this book, and to reach an understanding of all the ingredients that go into the one single definition of dagger compact categories would take more than a single chapter.
But it’s a good idea to be aware that there are researchers in quantum information science who work *entirely* from this approach, known as **categorical quantum mechanics**.

One particular method within this approach is the use of **string diagrams**, which allow for the use of so-called diagrammatic reasoning, with the **ZX-calculus** being a particularly successful example.
For an introduction to string diagrams of this flavour, it’s maybe a good idea to start with understanding how they can express the linear algebra that you already know.
For example, Pawel Sobocinski’s “Graphical Linear Algebra” aims to teach linear algebra entirely through the introduction of string diagrams.

This is an

*axiom*, justified by experimental evidence, and also by some sort of mathematical intuition. So, in this book, we take this as a*fact*that we do not question. It is, however, very interesting to question it:*why should we assume this to be true?*↩︎