Bra-Ket Notation

Bra-ket notation was introduced by Paul Dirac in 1939 as a compact and coordinate-independent way to express quantum mechanical operations. It is the universal language of quantum computing and quantum information theory.

Kets: State Vectors

A ket $|\psi\rangle$ represents a quantum state as a vector in a complex Hilbert space. For a single qubit, the computational basis states are:

• $|0\rangle$ corresponds to the column vector $(1, 0)^T$
• $|1\rangle$ corresponds to the column vector $(0, 1)^T$

A general qubit state is a superposition: $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$, where $\alpha$ and $\beta$ are complex amplitudes satisfying $|\alpha|^2 + |\beta|^2 = 1$.

For an $n$-qubit system, the ket lives in a $2^n$-dimensional Hilbert space. The computational basis consists of all $n$-bit strings: $|00\ldots0\rangle, |00\ldots1\rangle, \ldots, |11\ldots1\rangle$.

Bras: Dual Vectors

A bra $\langle\psi|$ is the conjugate transpose of the corresponding ket. If $|\psi\rangle = (\alpha, \beta)^T$, then $\langle\psi| = (\alpha^*, \beta^*)$ as a row vector. The operation of taking a ket to a bra is called the Hermitian conjugate or dagger operation: $\langle\psi| = (|\psi\rangle)^\dagger$.

Inner Products and Probabilities

The inner product of two states is written $\langle\phi|\psi\rangle$. It is a complex number computed as the matrix product of the bra row vector with the ket column vector.

For orthonormal basis states: $\langle0|0\rangle = \langle1|1\rangle = 1$ and $\langle0|1\rangle = \langle1|0\rangle = 0$.

The Born rule connects inner products to measurement probabilities. If you measure $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ in the computational basis:

• $P(\text{outcome } 0) = |\langle0|\psi\rangle|^2 = |\alpha|^2$
• $P(\text{outcome } 1) = |\langle1|\psi\rangle|^2 = |\beta|^2$

Outer Products and Operators

The outer product $|\psi\rangle\langle\phi|$ is a matrix (operator). It maps any state $|\chi\rangle$ to the state $(\langle\phi|\chi\rangle)\,|\psi\rangle$.

The projector onto state $|\psi\rangle$ is $P = |\psi\rangle\langle\psi|$. Any observable can be written in terms of its eigenvalues and projectors using the spectral decomposition: $O = \sum_i \lambda_i |i\rangle\langle i|$.

Tensor Products for Multi-Qubit Systems

For multi-qubit systems, states are combined with the tensor product: $|\psi_A\rangle \otimes |\psi_B\rangle$ is written $|\psi_A\rangle|\psi_B\rangle$ or $|\psi_A \psi_B\rangle$. The two-qubit computational basis is $|00\rangle, |01\rangle, |10\rangle, |11\rangle$, corresponding to the four basis vectors of the 4-dimensional tensor product space.

Common Conventions

In quantum computing, qubits are numbered from right to left in ket notation: in $|01\rangle$, qubit 0 is on the right and holds value 1, while qubit 1 is on the left and holds value 0. This convention matches the binary representation of integers but can be a source of confusion when comparing to circuit diagrams, where qubit 0 is typically drawn at the top.

See also

• Hilbert Space
• Superposition
• Born Rule
• Density Matrix