Normalization - Quantum States & Qubits Step by Step

Normalization checks that the squared probabilities for all outcomes close to one shared budget. The closure is exact only for the complete declared two-state basis and ideal amplitudes; real state reconstruction can miss leakage, suffer drift, and carry statistical plus systematic uncertainty.

A valid state has probabilities that add exactly to one. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Probabilities must add to one

The checked probabilities are 9/25 and 16/25.

\frac{9}{25} + \frac{16}{25} = 1

The amplitude pair is normalized

Because the squared amplitudes sum to 1, the state is normalized.

\lVert \psi \rVert^{2} = 1

Several exact amplitude pairs close

The lesson uses the first table row. Other exact rows show the same normalization rule: square both amplitudes, then add.

\begin{array}{c|c|c}a_{\text{zero}}&a_{\text{one}}&P_{\text{zero}}+P_{\text{one}}\\\frac{3}{5}&\frac{4}{5}&1\\\frac{5}{13}&\frac{12}{13}&1\\\frac{8}{17}&\frac{15}{17}&1\end{array}