Bloch Sphere Explorer

Relative phase is the core mechanism of quantum computing. On the equator of the Bloch sphere, the state is a superposition of |0⟩ and |1⟩. When measuring in the X or Y basis, these two components interfere. Depending on the phase phi, they either add constructively (aligning in the complex plane to yield 100% probability) or destructively (canceling to 0%). The interactive compass below shows this dynamic interference as delta varies.

A single qubit's entire state is just one point on this sphere's surface. The north pole is |0⟩, the south pole is |1⟩, and everything between them is a superposition. Latitude (θ) sets how the odds split between 0 and 1 — the poles are certainty, the equator is a 50/50 coin — while longitude (φ) sets the phase. The practical takeaway: measuring in the Z basis simply asks which hemisphere your point leans toward.

Notice the half-angle: the state uses cos(θ/2), not cos(θ). That factor of two is the fingerprint of a deep link — 3D rotations form the group SO(3), but a qubit's real state space is SU(2), which wraps around it twice. The consequence is startling: rotate a qubit a full 360° and its state vector flips to -|ψ⟩ instead of returning home; only a second turn, a full 720°, truly restores it. Every electron shares this spinor behavior, dramatized by Dirac's belt trick.