Blochsirkel

Blochsirkel, or the Bloch circle, is a two-dimensional representation used in quantum information to visualize the state of a single qubit. It is derived from the Bloch sphere, a three-dimensional geometrical representation of a qubit’s state. The Bloch circle is the projection of the Bloch sphere onto the equatorial plane and illustrates states with equal probability amplitudes for the basis states |0⟩ and |1⟩.

A pure qubit state can be written as |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩. Its Bloch vector is r = (sin θ cos φ, sin θ sin φ, cos θ). In the Bloch circle, θ = π/2, giving r = (cos φ, sin φ, 0). Thus each point on the unit circle corresponds to a pure state in this equatorial sector, with φ as the angle around the circle representing the relative phase between |0⟩ and |1⟩. States with θ ≠ π/2 or mixed states do not lie on the circle; their projection lies inside the Bloch disk.

In more general terms, the interior of the disk represents mixed states, and the distance from the origin reflects the degree of purity: pure states lie on the circumference (maximum radius), while mixed states lie closer to the center. The coordinates on the circle relate to the expectation values ⟨σ_x⟩ and ⟨σ_y⟩, while ⟨σ_z⟩ encodes the out-of-plane component.

The Bloch circle is a teaching and visualization aid for single-qubit operations: rotations around the x or y axis correspond to moves around the circle, and phase shifts correspond to rotation along its circumference. It complements the full Bloch sphere in accessible quantum computing explanations.

See also: Bloch sphere, qubit, Pauli matrices.