Blochsfär

Blochsfär, commonly called the Bloch sphere in English, is a geometrical representation of the quantum state of a two-level system, such as a spin-1/2 particle or a qubit. It provides an intuitive picture of superposition, coherence, and measurement by mapping states to points inside or on the surface of a unit sphere.

Any state of a two-level system can be written in terms of a density matrix ρ = 1/2 (I + r · σ), where I is the identity, σ = (σx, σy, σz) are the Pauli matrices, and r is a real three-dimensional Bloch vector. The vector r lies inside the unit ball: |r| ≤ 1. Pure states correspond to points on the surface with |r| = 1, while mixed states lie inside with |r| < 1. The components of r are often written as r_x, r_y, r_z, and can be parameterized by polar angles as r = (sin θ cos φ, sin θ sin φ, cos θ).

Geometrically, the north and south poles represent the eigenstates of σz (often denoted |0⟩ and |1⟩), while points on the equator represent equal superpositions with varying relative phase. Unitary evolution of the qubit corresponds to a rotation of the Bloch vector: r → R r, where R ∈ SO(3) is the rotation associated with the unitary on the state space via the SU(2)–SO(3) correspondence.

The Bloch sphere is a foundational tool in quantum information and quantum optics for visualizing qubits, performing state tomography, and illustrating concepts such as superposition and decoherence. It is specifically suited to two-level systems; higher-dimensional states admit generalized Bloch representations.