Orthonormal Bases

A basis for a vector space is a set of vectors from which every other vector in that space can be built as a linear combination. The computational basis $\{|0\rangle, |1\rangle\}$ is the default basis for a single qubit — but it carries an additional structure that makes quantum mechanics work out cleanly: it is orthonormal.

What orthonormality means

A set of vectors is orthonormal when two conditions hold simultaneously:

1. Normalization — every vector has unit length. For a ket, this means $\langle\psi|\psi\rangle = 1$.
2. Orthogonality — distinct vectors in the set are perpendicular. For two different kets $|\varphi\rangle$ and $|\psi\rangle$, this means $\langle\varphi|\psi\rangle = 0$.

Written compactly using the Kronecker delta $\delta_{ij}$ (which equals 1 when $i = j$ and 0 otherwise), the orthonormality condition for any basis $\{|e_i\rangle\}$ is

$\langle e_i | e_j \rangle = \delta_{ij}.$

Verifying $\{|0\rangle, |1\rangle\}$

The column-vector representations are

$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$

For real column vectors the inner product is just the dot product, so the four cases are:

All four results match the Kronecker-delta table: the diagonal entries are 1 and the off-diagonal entries are 0. The computational basis is therefore orthonormal.

Why it matters

Orthonormality is not a cosmetic detail — it is what lets the Born rule work. Because $\langle 0|1\rangle = 0$, the two measurement outcomes are mutually exclusive: a measurement collapses the state to exactly one of them. Because $\langle 0|0\rangle = \langle 1|1\rangle = 1$, the probabilities $|\langle 0|\psi\rangle|^2 + |\langle 1|\psi\rangle|^2 = 1$ always sum to one for any normalized $|\psi\rangle$. An orthonormal basis turns the abstract formalism into a consistent probability theory.

Try it

Compute all four inner products $\langle 0|0\rangle$, $\langle 0|1\rangle$, $\langle 1|0\rangle$, $\langle 1|1\rangle$ and return their sum. The orthonormality conditions determine exactly what number you should get.

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