Blochvector

The Bloch vector is a three-dimensional real vector used to represent the state of a two-level quantum system, or qubit, in a compact geometric form. A qubit state can be described by a density matrix ρ, and the Bloch vector r is defined by the relation ρ = 1/2 (I + r · σ), where σ = (σx, σy, σz) are the Pauli matrices and I is the identity. The components of r are given by r_i = Tr(ρ σ_i) = ⟨σ_i⟩, the expectation values of the Pauli operators.

The vector r lies inside the Bloch ball, a unit ball in R^3. Pure states correspond to Bloch vectors of unit length (|r| = 1) and thus lie on the surface of the Bloch sphere, while mixed states have |r| < 1. The Bloch sphere provides a geometric visualization of single-qubit states, with points on the surface representing coherent superpositions and the origin representing the maximally mixed state.

Under time evolution by a unitary operator U, the Bloch vector undergoes a rotation: r → r′ = R r, where R ∈ SO(3) is the rotation corresponding to U via the relation U σ U† = R σ. This makes the dynamics of single qubits resemble rigid-body rotations on the sphere.

Measurements along a unit vector n yield probabilities determined by the projection along that axis: the probability of obtaining the +1 outcome for a measurement along n is p = (1 + r · n)/2, and the expectation value along n is r · n.

The Bloch vector is specifically defined for two-level systems; higher-dimensional quantum systems use a generalized Bloch vector constructed from the generators of SU(d), with the qubit case using the Pauli matrix basis. The Bloch vector framework provides a compact, intuitive picture of state, measurement, and unitary evolution in quantum information.