Norm and Normalization

What is a norm?

Given a vector with $n$ components $v = (v_1, v_2, \ldots, v_n)$, its Euclidean norm (also called the $\ell^2$-norm or simply "the length") is

$\|v\| = \sqrt{\sum_{k=1}^{n} |v_k|^2}.$

The vertical bars $|v_k|^2$ are there for generality: when the components are complex numbers, $|v_k|^2 = v_k^* v_k$ is the squared modulus. For real components the expression reduces to the familiar Pythagorean sum.

Example. For $v = (1, 2, 2)$:

$\|v\| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3.$

Normalization

A vector is called normalized (or a unit vector) if its norm equals 1. To normalize $v$, divide every component by $\|v\|$:

$\hat{v} = \frac{v}{\|v\|} = \left(\frac{v_1}{\|v\|},\, \frac{v_2}{\|v\|},\, \ldots,\, \frac{v_n}{\|v\|}\right).$

One check: $\|\hat{v}\| = 1$ always, because

$\|\hat{v}\| = \frac{\|v\|}{\|v\|} = 1.$

Why quantum mechanics insists on normalization

In quantum mechanics a state is represented by a state vector (or ket) $|\psi\rangle$. The Born rule says the probability of finding the system in basis state $|k\rangle$ is $|\langle k|\psi\rangle|^2$ — the squared modulus of the corresponding amplitude. Probabilities must sum to 1:

$\sum_k |\langle k|\psi\rangle|^2 = 1 \quad\Longleftrightarrow\quad \|\,|\psi\rangle\,\| = 1.$

This is the physical reason normalization is not optional: an un-normalized state vector gives probabilities that do not add up to 1, which is physically meaningless. Whenever you construct a state vector from scratch — say, by superposing several basis states — the first thing to do is normalize it.

Worked example

Suppose the initial state of a three-level system is proportional to $(1, 2, 2)$. The norm is 3 (computed above), so the normalized state is

$|\psi\rangle = \left(\frac{1}{3},\, \frac{2}{3},\, \frac{2}{3}\right).$

Check: $\left(\frac{1}{3}\right)^2 + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^2 = \frac{1}{9} + \frac{4}{9} + \frac{4}{9} = \frac{9}{9} = 1$. The probabilities sum to 1, as required.

Try it

This is a numerical exercise — your code should return a number, not a circuit. Start from the state vector $(1, 2, 2)$, normalize it, and return its first component (which equals $\tfrac{1}{3}$).

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