BlochKugel

The Bloch Kugel, internationally known as the Bloch sphere, is a geometrical representation of the quantum state of a two-level system, especially a qubit. It maps density operators to points on or inside a unit sphere and is widely used in quantum information, nuclear magnetic resonance, and quantum optics.

A qubit state is described by a density matrix ρ with Tr(ρ) = 1. It can be written as ρ = 1/2 (I + r · σ), where σ = (σ_x, σ_y, σ_z) are the Pauli matrices and r is the Bloch vector with magnitude |r| ≤ 1. Pure states lie on the sphere surface (|r| = 1); mixed states lie inside the sphere; the completely mixed state is at r = 0, ρ = I/2. The north and south poles correspond to the eigenstates of σ_z, often written as |0⟩ and |1⟩.

Evolution: For closed systems, unitary evolution corresponds to rotations of the Bloch vector around an axis determined by the Hamiltonian. Decoherence and dissipation shrink the vector length toward the origin. The coordinates r_x, r_y, r_z equal Tr(ρ σ_x), Tr(ρ σ_y), Tr(ρ σ_z), respectively.

Applications and history: The Bloch Kugel was introduced by Felix Bloch. It provides an intuitive picture for single-qubit operations, state tomography, and gate visualization in quantum computing, as well as in magnetic resonance techniques.