Blochsphere

The Bloch sphere is a geometrical representation of the pure state space of a two-level quantum system, or qubit. Each pure state corresponds to a point on the surface of the unit sphere in three-dimensional real space. Mixed states lie inside the sphere. The concept traces back to Felix Bloch, who introduced it as a convenient way to visualize spin-1/2 states.

For a pure state in the computational basis {|0>, |1>}, the state can be parameterized by θ ∈ [0, π] and φ ∈ [0, 2π): |ψ> = cos(θ/2)|0> + e^{iφ} sin(θ/2)|1>. The corresponding Bloch vector r is (sin θ cos φ, sin θ sin φ, cos θ). The density matrix is ρ = (I + r·σ)/2, where σ = (σx, σy, σz) are the Pauli matrices.

Measurement along an axis a yields probability p = (1 + a·r)/2, with a a unit vector. Unitary evolution corresponds to rotation of r in three-dimensional space; a unitary U ∈ SU(2) induces a rotation R ∈ SO(3) such that r → R r.

The length of r encodes state purity: |r| = 1 for pure states and |r| < 1 for mixed states. Linear entropy can be expressed as S_L = 1 − |r|^2.

Beyond qubits, the Bloch representation generalizes to N-level systems with a Bloch vector in R^{N^2−1}. For N > 2 the state space is not a simple sphere, but the vector picture remains a useful visualization tool in quantum information and quantum computation.