Operators
A linear map between two vector spaces \mathcal{H} and \mathcal{K} is a function A\colon\mathcal{H}\to\mathcal{K} that respects linear combinations:
A(c_1|v_1\rangle+c_2|v_2\rangle)=c_1 A|v_1\rangle+c_2 A|v_2\rangle
for any vectors |v_1\rangle,|v_2\rangle and any complex numbers c_1,c_2.
We will focus mostly on endomorphisms, that is, maps from \mathcal{H} to \mathcal{H}, and we will call them operators.
The symbol \mathbf{1} is reserved for the identity operator that maps every element of \mathcal{H} to itself (i.e. \mathbf{1}|v\rangle=|v\rangle for all |v\rangle\in\mathcal{H}).
The product BA of two operators A and B is the operator obtained by first applying A to some ket |v\rangle and then B to the ket which results from applying A:
(BA)|v\rangle = B(A|v\rangle).
The order does matter: in general, BA\neq AB.
In the exceptional case in which AB=BA, one says that these two operators commute.
The inverse of A, written as A^{-1}, is the operator that satisfies AA^{-1}=\mathbf{1}=A^{-1}A.
For finite-dimensional spaces, one only needs to check 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 A is uniquely determined by the entries of its matrix: A_{ij}=\langle i|A|j\rangle.
The adjoint, or Hermitian conjugate, of an linear map A, denoted by A^\dagger, is defined by the relation
\begin{gathered}
\langle i|A^\dagger|j\rangle
= \langle j|A|i\rangle^\star
\\\text{for all $|i\rangle,|j\rangle\in\mathcal{H}$}
\end{gathered}
and \dagger turns (n\times m) matrices into (m\times n) matrices.
An operator A is said to be
- normal if AA^\dagger = A^\dagger A
- unitary if A^\dagger=A^{-1}
- Hermitian (or self-adjoint) if A^\dagger = A.
In particular then, being unitary implies being normal, since if A^\dagger=A^{-1} then AA^\dagger=A^\dagger A, since both of these are equal to \mathbf{1}.
Note also that unitary and Hermitian operators must indeed be 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.
Note that unitary operators preserve the inner product: given a unitary operator U and two kets |a\rangle and |b\rangle, and defining |a'\rangle=U|a\rangle and |b'\rangle=U|b\rangle, we have that
\begin{gathered}
\langle a'|=\langle a|U^\dagger
\\\langle b'|=\langle b|U^\dagger
\\\langle a'|b'\rangle=\langle a|U^\dagger U|b\rangle=\langle a|\mathbf{1}|b\rangle=\langle a|b\rangle.
\end{gathered}
Preserving the inner product implies preserving the norm induced by this product, i.e. unit state vectors are mapped to unit state vectors, i.e. 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.