Dirac notation — kets and bras

A ket is a column

$\lvert\psi\rangle$ means a state vector. For one qubit,

$$\lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle =\begin{pmatrix}\alpha\\\beta\end{pmatrix}.$$

Its corresponding bra is the conjugate-transposed row:

$$\langle\psi\rvert=(\alpha^*,\beta^*).$$

Complex conjugation matters. If $\beta=i/\sqrt2$, its conjugate is $-i/\sqrt2$.

Inner products

The expression $\langle\phi\vert\psi\rangle$ is a complex number. It describes overlap between vectors. The basis states are orthonormal: $\langle0\vert0\rangle=1$ and $\langle0\vert1\rangle=0$.

For $\lvert\psi\rangle=(\lvert0\rangle+i\lvert1\rangle)/\sqrt2$,

$$\langle\psi\vert\psi\rangle=\frac12+\frac{(-i)i}{2}=1.$$

The probability of a Z-basis outcome 1 is $|\langle1\vert\psi\rangle|^2=1/2$.

Outer products are operators

$\lvert0\rangle\langle0\rvert$ is a matrix, not a scalar:

$$\lvert0\rangle\langle0\rvert=\begin{pmatrix}1&0\\0&0\end{pmatrix}.$$

It projects onto the $\lvert0\rangle$ direction. The distinction is visual: a bra followed by a ket gives an inner product; a ket followed by a bra gives an operator.

Check yourself: calculate $\langle+\vert-\rangle$. The two terms cancel, giving zero, even though their Z-basis probabilities are identical.

Reference: IBM Quantum — Single systems.