Bras and Inner Products

Dirac notation pairs every ket $|\psi\rangle$ with a bra $\langle\psi|$. If the ket is a column vector, the bra is its conjugate transpose — a row vector whose entries are the complex conjugates of the ket's entries. Together they give a compact way to write the inner product of two quantum states.

From kets to bras

For a general single-qubit ket $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, the column-vector form is

$|\psi\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}.$

The corresponding bra is

$\langle\psi| = \begin{pmatrix} \alpha^* & \beta^* \end{pmatrix},$

where $\alpha^*$ denotes the complex conjugate of $\alpha$. When the amplitudes are real — as they are for the states we will study first — conjugation does nothing and the bra is just the row version of the ket.

The inner product

The inner product $\langle\varphi|\psi\rangle$ is defined by multiplying the row vector $\langle\varphi|$ by the column vector $|\psi\rangle$ in the usual matrix-multiplication sense:

$\langle\varphi|\psi\rangle = \sum_i \varphi_i^* \, \psi_i.$

Two important properties follow directly from this definition:

• Normalization: $\langle\psi|\psi\rangle = 1$ for any physical (unit-norm) state. This is just the statement that the probabilities sum to one.
• Orthogonality of basis states: $\langle 0|1\rangle = 0$ and $\langle 1|0\rangle = 0$, while $\langle 0|0\rangle = \langle 1|1\rangle = 1$.

A worked example

The state $|{+}\rangle = \tfrac{1}{\sqrt{2}}|0\rangle + \tfrac{1}{\sqrt{2}}|1\rangle$ has the column representation

$|{+}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}.$

Its bra is therefore $\langle{+}| = \tfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \end{pmatrix}$.

The inner product with $|0\rangle = \binom{1}{0}$ is

$\langle{+}|0\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{1}{\sqrt{2}} \cdot 1 + \frac{1}{\sqrt{2}} \cdot 0 = \frac{1}{\sqrt{2}}.$

Squaring the magnitude gives $|\langle{+}|0\rangle|^2 = \tfrac{1}{2}$, which is exactly the 50 % chance of measuring 0 in the state $|{+}\rangle$ — confirming that the inner product carries the measurement probability we already know from the Born rule.

Try it

Compute the numerical value of $\langle{+}|0\rangle$ and return it.

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