Blochvektorn

Blochvektorn, or the Bloch vector, is a three-dimensional real vector used to describe the state of a two-level quantum system, such as a qubit, in the Bloch sphere representation. Any state can be represented by a density matrix ρ, which can be written as ρ = 1/2 (I + r · σ), where σ = (σx, σy, σz) are the Pauli matrices and r is the Bloch vector with components (rx, ry, rz). The magnitude |r| ≤ 1, with pure states having |r| = 1 and mixed states having |r| < 1. The coordinates are given by r_i = Tr(ρ σ_i) and also equal to the expectation values ⟨σ_i⟩.

In the Bloch sphere picture, the state space of a two-level system is represented by the unit ball in R^3, with pure states on the surface. Operations on the state correspond to rotations and contractions of the Bloch vector: unitary evolution under a Hamiltonian H = (ħ/2) ω · σ leads to dr/dt = ω × r, i.e., precession around the axis defined by ω. In the presence of dissipation and decoherence, the Bloch equations generalize to include relaxation times T1 and T2, causing the vector to contract toward the Bloch ball center.

Measurement of Pauli operators yields components of the Bloch vector: ⟨σ_i⟩ = r_i. The Bloch vector provides an intuitive geometric visualization of quantum states and their dynamics, and is widely used in quantum information science, nuclear magnetic resonance, and quantum optics.

Historically, the representation is named after Felix Bloch, who developed the formalism for spin systems in the mid-20th century.