Blochball

Blochball is a term used to describe the geometric representation of the state space of a single qubit in quantum information theory. It is closely related to the better-known Bloch sphere, but the ball terminology emphasizes that the interior also represents valid quantum states.

In the standard formulation, any qubit state can be described by a density matrix ρ. This state is mapped to a three-dimensional real vector r = (x, y, z) inside the unit ball, via ρ = 1/2 (I + r · σ), where σ = (σx, σy, σz) are the Pauli matrices and I is the identity. The condition ρ ≥ 0 and Tr(ρ) = 1 translates to the constraint ||r|| ≤ 1, so the set of all valid states forms the Bloch ball of radius 1.

Geometry and state types: Points on the surface of the ball (||r|| = 1) correspond to pure states, while interior points (||r|| < 1) represent mixed states. The length of r reflects the state's purity, with shorter vectors indicating greater mixing. The direction of r encodes the expectation values of the Pauli observables along x, y, and z axes.

Dynamics and measurements: Unitary evolution of the qubit corresponds to rotations of the Bloch ball, i.e., the vector r is rotated without changing its magnitude. Non-unitary processes, such as decoherence or dissipation, generally move r toward the origin, increasing mixedness. Measurement along a direction n yields probabilities (1 ± r · n)/2 for the possible outcomes, consistent with quantum theory.

Applications and notes: The Blochball provides an intuitive visualization for single-qubit states, quantum state tomography, and simple illustrations of quantum operations. It is the three-dimensional representation of the convex set of all density operators for a two-level system; higher-dimensional generalizations exist but do not form a ball in ordinary three-space.