3D Bloch Sphere: Qubit Superposition & Decoherence

Bloch Sphere

Initial polar angle θ 60°

Larmor precession ω 1.2 rad/s

Environment coupling γ 0.15

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P(|0⟩)

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P(|1⟩)

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|r_xy| (coherence)

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Purity Tr(ρ²)

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Measurements

Outcome tally

0 / 0

How it works

Every pure single-qubit state maps to a point on the unit Bloch sphere: (x,y,z) = (sinθ cosφ, sinθ sinφ, cosθ), where z encodes the population split (z=+1 is |0⟩, z=−1 is |1⟩) and the azimuth φ carries the relative phase between the two amplitudes. Free evolution just spins the vector around the pole at rate ω — Larmor precession. Coupling to the environment doesn't touch z (the populations are untouched), but it randomizes φ, so the vector's equatorial radius r_xy shrinks exponentially toward the axis. A vector sitting exactly on the axis, inside the sphere, is a classical probabilistic mixture of |0⟩ and |1⟩ — no interference left, only odds.

z(t) = cosθ₀                (unchanged by dephasing)
r_xy(t) = sinθ₀ · e^(−γt)    (shrinks toward the axis)
P(|0⟩) = (1 + z)/2           (Born rule, Bloch form)

θ — initial angle from the north pole; θ=90° starts the vector on the equator, an even superposition.
ω — precession rate around the pole; a real qubit's energy gap between |0⟩ and |1⟩ sets this.
γ — decoherence rate; watch the fading trail spiral inward from the sphere's surface toward the axis as coherence is lost.
Measure — samples an outcome via the Born rule and snaps the vector to whichever pole was drawn, resetting it to a pure state.

The full-length vector on the surface is a pure superposition capable of interference; the shrunken vector near the axis has the same measurement odds but has irreversibly lost that capability — the geometric picture of how "quantum" fades into "classical."

State vector |0⟩ pole |1⟩ pole Trajectory trail

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