Normalization

A qubit's state is a vector in a two-dimensional complex vector space. We write it as

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,$

where $\alpha$ and $\beta$ are complex numbers called probability amplitudes. The Born rule says that if you measure the qubit, you get outcome $0$ with probability $|\alpha|^2$ and outcome $1$ with probability $|\beta|^2$ .

Because probabilities must add to one, the amplitudes are constrained:

$|\alpha|^2 + |\beta|^2 = 1.$

This is the normalization condition. Geometrically, it means the state vector lies on the unit sphere in the two-dimensional complex Hilbert space (a three-sphere $S^3$ when viewed as a subset of $\mathbb{R}^4$ ). The familiar Bloch sphere ( $S^2$ ) is the quotient of that $S^3$ by global phase: two normalized states that differ only by an overall factor $e^{i\phi}$ are physically identical, and collapsing those equivalence classes maps $S^3$ onto the real two-sphere $S^2$ .

Why normalization matters

Every legal qubit state satisfies the normalization condition exactly. Any vector that does not satisfy it cannot represent a physical state, because the predicted probabilities would not sum to one — a logical impossibility. Normalization is therefore the first sanity check to run on any purported qubit state.

Checking norm in code

In practice you compute the norm squared

$\|\psi\|^2 = |\alpha|^2 + |\beta|^2$

and check whether it equals $1$ within a small numerical tolerance. Floating-point arithmetic introduces rounding errors of order $10^{-16}$ , so a tolerance of $10^{-9}$ gives a safe margin while still catching states that are genuinely un-normalized.

For example, the equal-superposition state $|{+}\rangle = \tfrac{1}{\sqrt{2}}|0\rangle + \tfrac{1}{\sqrt{2}}|1\rangle$ has

$|\alpha|^2 + |\beta|^2 = \frac{1}{2} + \frac{1}{2} = 1,$

so it is correctly normalized.

Try it

Complete the function below to verify that the given amplitudes describe a normalized qubit state. Return true if the norm squared is within $10^{-9}$ of $1$ , and false otherwise.

Run your code to see the quantum state.