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> represents a quantum state as a vector in a complex Hilbert space. For a single qubit, the computational basis states are:

• |0> corresponds to the column vector (1, 0)^T
• |1> corresponds to the column vector (0, 1)^T

A general qubit state is a superposition: |psi> = alpha|0> + beta|1>, 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…0>, |00…1>, …, |11…1>.

Bras: Dual Vectors

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

Inner Products and Probabilities

The inner product of two states is written <phi|psi>. 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 and <0|1> = <1|0> = 0.

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

• P(outcome 0) = |<0|psi>|^2 = |alpha|^2
• P(outcome 1) = |<1|psi>|^2 = |beta|^2

Outer Products and Operators

The outer product |psi><phi| is a matrix (operator). It maps any state |chi> to the state (<phi|chi>) * |psi>.

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

Tensor Products for Multi-Qubit Systems

For multi-qubit systems, states are combined with the tensor product: |psi_A> tensor |psi_B> is written |psi_A>|psi_B> or |psi_A psi_B>. The two-qubit computational basis is |00>, |01>, |10>, |11>, 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>, 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