Kets: |ψ⟩ Notation

Every quantum state is written using a compact notation invented by Paul Dirac. The symbol $|\psi\rangle$ is called a ket — it is simply a label for a state vector. For a single qubit, the two standard building blocks are $|0\rangle$ and $|1\rangle$ , called the computational basis states.

Kets as column vectors

Behind the label is an ordinary column vector. The correspondence is

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

Any single-qubit state $|\psi\rangle$ is a superposition — a linear combination with complex coefficients called amplitudes:

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix},

where $\alpha, \beta \in \mathbb{C}$ and the normalization condition $|\alpha|^2 + |\beta|^2 = 1$ must hold so that measurement probabilities sum to one.

Reading off probabilities

Once you have a ket, the Born rule turns amplitudes into probabilities. The probability of observing outcome $b \in \{0, 1\}$ when measuring in the computational basis is

P(b) = |\langle b | \psi \rangle|^2 = |\text{amplitude of } |b\rangle|^2.

For example, consider the normalized state

|\psi\rangle = \frac{\sqrt{3}}{2}|0\rangle + \frac{1}{2}|1\rangle = \begin{pmatrix} \tfrac{\sqrt{3}}{2} \\ \tfrac{1}{2} \end{pmatrix}.

Reading off the amplitudes: $\alpha = \tfrac{\sqrt{3}}{2}$ and $\beta = \tfrac{1}{2}$ . The probabilities are

P(0) = \left|\frac{\sqrt{3}}{2}\right|^2 = \frac{3}{4}, \qquad P(1) = \left|\frac{1}{2}\right|^2 = \frac{1}{4}.

These sum to one, confirming the state is normalized: $\tfrac{3}{4} + \tfrac{1}{4} = 1$ .

Try it

Apply the Born rule to the state $|\psi\rangle = \tfrac{\sqrt{3}}{2}|0\rangle + \tfrac{1}{2}|1\rangle$ and return $P(1)$ , the probability of measuring outcome $1$ .

Run your code to see the quantum state.