Probability Amplitude

quantum-computing

Definition

Probability Amplitude

Let $\mathcal{H}$ be a complex Hilbert space with orthonormal basis $B=\{|b_i⟩{\}}_{i\in I}$. For a normalised quantum state $|\psi ⟩\in \mathcal{H}$,

$$|\psi ⟩=\sum _{i\in I}{\alpha }_i|b_i⟩,{\alpha }_i=⟨b_i|\psi ⟩\in \mathbb{C}.$$

The complex coefficient ${\alpha }_i$ is the probability amplitude of the basis vector $|b_i⟩$ in $|\psi ⟩$. Under measurement in the basis $B$, the probability of outcome $b_i$ is

$$\mathrm{Pr}(b_i\mid \psi )=|{\alpha }_i{|}^2.$$

Thus a probability amplitude is a complex coordinate of a quantum state, and the Born rule maps it to a classical probability by squared modulus.

Projection

Given a single qubit state

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩\text{with }\alpha ,\beta \in \mathbb{C}\wedge |\alpha {|}^2+|\beta {|}^2=1.$$

The inner product is sesquilinear, i.e., it takes the ket as the linear side, thus:

$$\begin{array}{rl}⟨x|\psi ⟩& =⟨x|(\alpha |0⟩+\beta |1⟩)⟩\\ & =⟨x|\alpha |0⟩+⟨x|\beta |1⟩\\ & =\alpha ⟨x|0⟩+\beta ⟨x|1⟩\end{array}$$

Given that $\{|0⟩,|1⟩\}$ is orthonormal, i.e.

$$\begin{array}{rl}⟨0|0⟩& =⟨1|1⟩=1\\ ⟨0|1⟩& =⟨1|0⟩=0,\end{array}$$

substituting into the expression above gives

$$\begin{array}{rl}⟨0|\psi ⟩& =\alpha ⟨0|0⟩+\beta ⟨0|1⟩=\alpha \cdot 1+\beta \cdot 0=\alpha ,\\ ⟨1|\psi ⟩& =\alpha ⟨1|0⟩+\beta ⟨1|1⟩=\alpha \cdot 0+\beta \cdot 1=\beta .\end{array}$$

Thus the probability amplitude of a basis state $|x⟩$ equals its inner product with $|\psi ⟩$:

$${\alpha }_x=⟨x|\psi ⟩.$$