Normalised Quantum State

quantum-computing quantum-mechanics

Definition

Normalised Quantum State

Let $\mathcal{H}$ be a complex Hilbert space. A normalised quantum state is a quantum state represented by a vector $|\psi ⟩\in \mathcal{H}$ with unit norm:

$$\Vert |\psi ⟩\Vert =1.$$

Equivalently, using bra-ket notation,

$$⟨\psi |\psi ⟩=1.$$

Thus a normalised quantum state is a unit vector in the quantum state space.

Why Normalisation Is Required

The Born rule assigns probabilities by squared amplitudes. For a basis expansion

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

the total probability is

$$\sum _{i\in I}\mathrm{Pr}(b_i\mid \psi )=\sum _{i\in I}|{\alpha }_i{|}^2=⟨\psi |\psi ⟩.$$

Therefore probabilities sum to $1$ exactly when

$$⟨\psi |\psi ⟩=1.$$