Essential Mathematics for Quantum Computing

Appendix 1: Bra–ket Notation

Bra-ket notation was introduced by Paul Dirac in 1939 and is sometimes called Dirac notation consequently. Kets are denoted by a pipe ("|") and right-angle bracket ("⟩"), like so – |Label⟩, while bras are denoted by a left-angle bracket ("⟨") and pipe ("|"), like so ⟨Label|. Kets represent vectors in a Hilbert space and bras represent their covectors in a dual Hilbert space. The labels for kets and bras can be lowercase letters, numbers, and greek letters. Uppercase letters are usually reserved for operators, which we will get to later.

The computational basis vectors are represented by |0⟩ and |1⟩. It is important to note that the zero vector is denoted by 0 and is totally different than |0⟩. The zero vector, sometimes ...