Dirac Bra-Ket Notation

Dirac Bra-Ket Notation#

The mathematical language of quantum mechanics: kets \(|\psi\rangle\), bras \(\langle\phi|\), inner products, outer products, and their physical interpretation.

Dirac notation (bra-ket notation) is the standard language of quantum mechanics and quantum computing. Introduced by Paul Dirac in 1939, it provides a concise, basis-independent way to write quantum states, operators, and measurements. Every equation in quantum computing papers uses this notation. Learning it is not optional.

Kets: Quantum States#

A ket \(|\psi\rangle\) represents a column vector in a complex Hilbert space. For a qubit:

\[\begin{split}|\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}\end{split}\]

where \(\alpha, \beta \in \mathbb{C}\) and \(|\alpha|^2 + |\beta|^2 = 1\) (normalization).

The computational basis for a single qubit is:

\[\begin{split}|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\end{split}\]

Other commonly used single-qubit states:

State	Notation	Vector	Basis
Plus	\(\|+\rangle\)	\(\frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}\)	Hadamard (\(X\)) basis
Minus	\(\|-\rangle\)	\(\frac{1}{\sqrt{2}}\begin{pmatrix}1\\-1\end{pmatrix}\)	Hadamard (\(X\)) basis
Plus-\(i\)	\(\|+i\rangle\)	\(\frac{1}{\sqrt{2}}\begin{pmatrix}1\\i\end{pmatrix}\)	Circular (\(Y\)) basis
Minus-\(i\)	\(\|-i\rangle\)	\(\frac{1}{\sqrt{2}}\begin{pmatrix}1\\-i\end{pmatrix}\)	Circular (\(Y\)) basis

Bras: Dual Vectors#

A bra \(\langle\phi|\) is the conjugate transpose (Hermitian adjoint) of the corresponding ket:

\[\begin{split}\text{If } |\phi\rangle = \begin{pmatrix} a \\ b \end{pmatrix}, \text{ then } \langle\phi| = \begin{pmatrix} a^* & b^* \end{pmatrix}\end{split}\]

The bra lives in the dual space, the space of linear functionals on the Hilbert space.

Inner Products#

The inner product \(\langle\phi|\psi\rangle\) (a “bracket,” hence “bra-ket”) is a complex number obtained by multiplying a bra by a ket:

\[\begin{split}\langle\phi|\psi\rangle = \begin{pmatrix} a^* & b^* \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = a^*\alpha + b^*\beta\end{split}\]

Properties:

Conjugate symmetry: \(\langle\phi|\psi\rangle = \langle\psi|\phi\rangle^*\)
Linearity in the second argument: \(\langle\phi|(\alpha|\psi_1\rangle + \beta|\psi_2\rangle) = \alpha\langle\phi|\psi_1\rangle + \beta\langle\phi|\psi_2\rangle\)
Positive definiteness: \(\langle\psi|\psi\rangle \geq 0\), with equality only if \(|\psi\rangle = 0\)

Orthonormality#

Two states are orthogonal if \(\langle\phi|\psi\rangle = 0\). A state is normalized if \(\langle\psi|\psi\rangle = 1\). The computational basis is orthonormal:

\[\langle 0|0\rangle = 1, \quad \langle 1|1\rangle = 1, \quad \langle 0|1\rangle = 0, \quad \langle 1|0\rangle = 0\]

This can be written compactly as \(\langle i|j\rangle = \delta_{ij}\) (the Kronecker delta).

Physical Meaning#

The inner product \(\langle\phi|\psi\rangle\) is the probability amplitude for measuring state \(|\psi\rangle\) and finding it in state \(|\phi\rangle\). The probability is the squared modulus:

\[P = |\langle\phi|\psi\rangle|^2\]

This is the Born rule in Dirac notation.

Outer Products#

The outer product \(|\psi\rangle\langle\phi|\) is an operator (a matrix) that maps kets to kets:

In matrix form for qubits:

\[\begin{split}|0\rangle\langle 0| = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad |1\rangle\langle 1| = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\end{split}\]

\[\begin{split}|0\rangle\langle 1| = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad |1\rangle\langle 0| = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\end{split}\]

Projection Operators#

The outer product \(P_k = |k\rangle\langle k|\) is a projector onto state \(|k\rangle\). Projectors satisfy \(P_k^2 = P_k\) and \(P_k^\dagger = P_k\).

Completeness Relation#

For any orthonormal basis \(\{|k\rangle\}\), the projectors sum to the identity:

\[\sum_k |k\rangle\langle k| = I\]

This is called the completeness relation (or resolution of the identity). It is used constantly in quantum mechanics to insert bases, change representations, and decompose operators.

Operators in Dirac Notation#

Any linear operator \(A\) can be written using outer products:

\[A = \sum_{i,j} A_{ij} |i\rangle\langle j|\]

where \(A_{ij} = \langle i|A|j\rangle\) are the matrix elements of \(A\) in the basis \(\{|i\rangle\}\).

Expectation Values#

The expected value of an observable \(A\) when measured on state \(|\psi\rangle\):

where \(a_k\) are the eigenvalues and \(|a_k\rangle\) the eigenstates of \(A\).

Tensor Products#

\[\begin{split}|0\rangle \otimes |1\rangle = |0\rangle|1\rangle = |01\rangle = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}\end{split}\]

The four computational basis states for two qubits are:

\[\begin{split}|00\rangle = \begin{pmatrix}1\\0\\0\\0\end{pmatrix}, \; |01\rangle = \begin{pmatrix}0\\1\\0\\0\end{pmatrix}, \; |10\rangle = \begin{pmatrix}0\\0\\1\\0\end{pmatrix}, \; |11\rangle = \begin{pmatrix}0\\0\\0\\1\end{pmatrix}\end{split}\]

Notation Shortcuts

In quantum computing literature, you will see all of the following used interchangeably: \(|0\rangle \otimes |1\rangle = |0\rangle|1\rangle = |0,1\rangle = |01\rangle = |1\rangle_{\text{decimal}}\). The last form uses the decimal representation of the binary string.

Summary of Dirac Notation#

Symbol	Name	Type	Dimension (qubit)
\(\|\psi\rangle\)	Ket	Column vector	\(2 \times 1\)
\(\langle\psi\|\)	Bra	Row vector	\(1 \times 2\)
\(\langle\phi\|\psi\rangle\)	Inner product	Scalar (\(\mathbb{C}\))	\(1 \times 1\)
\(\|\psi\rangle\langle\phi\|\)	Outer product	Operator (matrix)	\(2 \times 2\)
\(\langle\psi\|A\|\psi\rangle\)	Expectation value	Scalar (\(\mathbb{R}\))	\(1 \times 1\)