Quantum States

Definition: Quantum Register

A quantum register is a sequence of qubits.

Definition: Quantum State

The state of a quantum register with $n$ qubits $|q_{n-1}⟩,\dots ,|q_0⟩$ or $⟨q_{n-1}|,\dots ,⟨q_0|$ is their Kronecker product, respectively:

$$|q_{n-1}⟩\otimes \dots \otimes |q_0⟩\text{or}⟨q_{n-1}|\otimes \dots \otimes ⟨q_0|$$

NOTATION

We use some shorthand notations for such products:

$$\begin{array}{rl}|q_{n-1}⟩|q_{n-2}⟩\cdots |q_0⟩& ⟨q_{n-1}|⟨q_{n-2}|\cdots ⟨q_0|\\ |q_{n-1},q_{n-2},\dots ,q_0⟩& ⟨q_{n-1},q_{n-2},\dots ,q_0|\\ |q_{n-1}{q}_{n-2}\cdots q_0⟩& ⟨q_{n-1}{q}_{n-2}\cdots q_0|\end{array}$$

The state of an $n$-qubit quantum register is a complex vector of dimension $2^n$.

NOTATION

The most common label for quantum states is the Greek letter $\psi$:

$$⟨\psi ||\psi ⟩$$