## 0.8 Outer products

Apart from the inner product **outer product**

- The result of
|u\rangle\langle v| acting on a ket|x\rangle is|u\rangle\langle v|x\rangle , i.e. the vector|u\rangle multiplied by the complex number\langle v|x\rangle . - Similarly, the result of
|u\rangle\langle v| acting on a bra\langle y| is\langle y|u\rangle\langle v| , i.e. the linear functional\langle v| multiplied by the complex number\langle y|u\rangle .

The product of two maps,

Any operator on *also* form an orthonormal basis then the operator simply “rotates” one orthonormal basis into another.
These are unitary operators which preserve the inner product.
In particular, if each *any* orthonormal basis, and it is one of the most ubiquitous and useful formulas in quantum theory, known as **completeness**.^{16}
For example, for any vector

Finally, note that calculating the adjoint of an outer product boils down to just swapping the order:

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**.

Not to be confused with “completeness” in the sense of Hilbert spaces.↩︎