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A qubit's true home is a complex Hilbert space, ℂ². We flatten it onto a friendlier picture — the Bloch sphere — by throwing away one real number: the global phase. Drag the dial below and watch what happens to the thing we threw away.
Notice the left dial (what you'd measure) repeats every 360°. The right dial — the actual state vector — only comes home every 720°. At 360° it points backwards: the state has picked up an overall minus sign that no measurement on the Bloch sphere can ever detect.
A pure qubit state is a unit vector in ℂ²:
Write out α and β as complex numbers and you have four real parameters. Normalization removes one. And then there's a second, quieter constraint: for any real number γ, the state eiγ|ψ⟩ predicts exactly the same outcome for every possible measurement as |ψ⟩ does. Global phase carries zero physical information. Remove that degree of freedom too and you're left with two real numbers — which is exactly enough to specify a point on a 2-sphere:
That's the Bloch sphere. It is a completely faithful map of every measurable fact about the qubit. It is not a faithful map of the state itself — it's the state with one axis of information, the global phase, projected away. Formally this is the Hopf fibration: the space of normalized states is a 3-sphere S³ sitting inside ℂ², and the map that discards global phase sends S³ onto S² two-to-one along circular fibers.
Here's the detail that makes the dial on the right spin at half speed. A physical rotation of a spin-½ particle by angle θ around an axis n̂ acts on the state vector as
that factor of /2 is not a typo or a convention someone could have dropped — it's forced by the algebra of the Pauli matrices, and it's the same θ/2 sitting inside the Bloch parametrization above. Since n̂·σ has eigenvalues ±1, plugging in θ = 2π gives
A full physical turn multiplies the state by −1. Only at θ = 4π does the exponent become a multiple of 2πi and R(4π) = +I return the state exactly to itself. Concretely, if you start at the Bloch point (0,1,0) — state |ψ₀⟩ = (|0⟩+i|1⟩)/√2 — and rotate about the axis facing you, the amplitudes trace out
which is exactly what's driving the readout above. Set θ=2π and you'll find α = −1/√2, β = −i/√2: precisely −|ψ₀⟩.
In 1975 two independent groups confirmed this with neutron interferometry: a single beam of neutrons is coherently split by a perfect silicon crystal, one path carries the spin through a magnetic field that rotates it, and the two paths are recombined. Because interference depends on the relative phase between the two paths, the invisible-to-the-Bloch-sphere minus sign becomes very visible: the recombined intensity oscillates with the field strength, and the interference pattern only returns to its starting point after a full 720° of spin rotation, not 360°. It's one of the cleanest experimental confirmations that the wavefunction, not just our description of it, genuinely lives one level of structure above the space of outcomes.
The general statement is that spatial rotations form the group SO(3), but the objects that transform spin-½ states form SU(2) — and the map SU(2) → SO(3) is two-to-one, with both U and −U in SU(2) mapping to the same rotation in SO(3). Every fermion inherits this: it's the reason spinors, unlike ordinary vectors, need a "square root of a rotation" to be described completely, and it's the same topological fact (π₁(SO(3)) = ℤ₂) that shows up again in the Hopf fibration connecting S³ to S². The Bloch sphere and the neutron interferometer are two views of one piece of topology.