Blochvektor

The Bloch vector is a real three-dimensional vector that provides a geometric representation of the state of a two-level quantum system, or qubit, through the Bloch sphere. It is named after Felix Bloch, whose work in the mid-20th century laid the foundations for this visualization used in quantum information and magnetic resonance.

For a quantum state described by the density operator ρ, the Bloch vector r has components r_i = Tr(ρ σ_i), where σ_x, σ_y, σ_z are the Pauli matrices. The density operator can be written as ρ = 1/2 (I + r · σ). The magnitude of r satisfies |r| ≤ 1, with equality for pure states and |r| < 1 for mixed states.

Geometrically, the Bloch sphere is the unit sphere in R^3. Pure states correspond to points on the surface, representing definite spin orientations, while mixed states lie inside the sphere. The north and south poles typically represent the eigenstates of σ_z, with other directions corresponding to superpositions.

Dynamics on the Bloch sphere reflect quantum evolution. Unitary evolution generated by a Hamiltonian H = (ħ/2) ω · σ induces a rotation of the Bloch vector around the axis set by ω. In open systems, decoherence and relaxation shrink the Bloch vector toward the origin, reflecting loss of purity.

Measurements of Pauli observables along a given axis yield the corresponding components of r, r_i = Tr(ρ σ_i). The Bloch vector thus provides an intuitive, compact picture of single-qubit states, their transformations, and their measurement statistics.

Applications of the Bloch vector span quantum information, quantum computing for visualizing single-qubit gates, state tomography, and areas such as NMR and quantum optics.