• Mathematics
  • Also: Dirac notation

Bra-Ket Notation

Bra-ket notation (Dirac notation) is a standard mathematical notation for quantum states where ket |psi> represents a column vector state, bra <psi| its conjugate transpose, and their product <phi|psi> the inner product.

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

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

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

For an nn-qubit system, the ket lives in a 2n2^n-dimensional Hilbert space. The computational basis consists of all nn-bit strings: ∣00…0⟩,∣00…1⟩,…,∣11…1⟩|00\ldots0\rangle, |00\ldots1\rangle, \ldots, |11\ldots1\rangle.

Bras: Dual Vectors

A bra ⟨ψ∣\langle\psi| is the conjugate transpose of the corresponding ket. If ∣ψ⟩=(α,β)T|\psi\rangle = (\alpha, \beta)^T, then ⟨ψ∣=(α∗,β∗)\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: ⟨ψ∣=(∣ψ⟩)†\langle\psi| = (|\psi\rangle)^\dagger.

Inner Products and Probabilities

The inner product of two states is written ⟨ϕ∣ψ⟩\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: ⟨0∣0⟩=⟨1∣1⟩=1\langle0|0\rangle = \langle1|1\rangle = 1 and ⟨0∣1⟩=⟨1∣0⟩=0\langle0|1\rangle = \langle1|0\rangle = 0.

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

  • P(outcome 0)=∣⟨0∣ψ⟩∣2=∣α∣2P(\text{outcome } 0) = |\langle0|\psi\rangle|^2 = |\alpha|^2
  • P(outcome 1)=∣⟨1∣ψ⟩∣2=∣β∣2P(\text{outcome } 1) = |\langle1|\psi\rangle|^2 = |\beta|^2

Outer Products and Operators

The outer product ∣ψ⟩⟨ϕ∣|\psi\rangle\langle\phi| is a matrix (operator). It maps any state ∣χ⟩|\chi\rangle to the state (⟨ϕ∣χ⟩) ∣ψ⟩(\langle\phi|\chi\rangle)\,|\psi\rangle.

The projector onto state ∣ψ⟩|\psi\rangle is P=∣ψ⟩⟨ψ∣P = |\psi\rangle\langle\psi|. Any observable can be written in terms of its eigenvalues and projectors using the spectral decomposition: O=∑iλi∣i⟩⟨i∣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: ∣ψA⟩⊗∣ψB⟩|\psi_A\rangle \otimes |\psi_B\rangle is written ∣ψA⟩∣ψB⟩|\psi_A\rangle|\psi_B\rangle or ∣ψAψB⟩|\psi_A \psi_B\rangle. The two-qubit computational basis is ∣00⟩,∣01⟩,∣10⟩,∣11⟩|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⟩|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