The Bloch Sphere
Picturing every possible single-qubit state as a point on a sphere
By the end of this topic you'll be able to
A qubit's state has two complex amplitudes — four real numbers — but normalization removes one degree of freedom, and the overall phase of the state is unobservable: |ψ⟩ and e^{iφ}|ψ⟩ give identical measurement statistics for every conceivable measurement. Strip away those two, and exactly two real parameters remain — enough to locate a single point on a sphere.
Binney and Skinner derive the general form directly from the requirement that a measurement along an arbitrary direction n, with polar coordinates (θ,φ), return a definite value: the state certain to give +½ along n works out to a specific combination of sin(θ/2) and cos(θ/2), each carrying its own phase factor. Written in the computing convention as |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩, θ is the polar angle from the north pole and φ the azimuthal angle around the equator.
|0⟩ occupies the north pole (θ=0) and |1⟩ the south pole (θ=π). The entire equator (θ=π/2) is filled with equal-superposition states distinguished purely by phase: |+⟩ sits at φ=0, |−⟩ at φ=π, and every other phase gives another point around the same circle.
A calculation of exactly this kind is what gave Binney and Skinner their result for a spin prepared along z and then measured along a tilted direction n: the probability of the opposite outcome comes out to sin²(θ/2), where θ is the angle between z and n. It vanishes when the two directions coincide and rises to one half when they are perpendicular — precisely the geometric statement that a measurement's outcome probabilities depend only on the angle between preparation and measurement axes.
This geometry makes the earlier claim that 'phase is invisible to direct measurement' visually obvious: a measurement in the computational basis reads off only θ, the polar angle, never φ. Two states that share a value of θ but differ in φ are measured identically in that basis, even though they are different states in every other respect.
That difference stops being hidden the instant a phase-sensitive gate is applied before measurement — which is exactly the mechanism behind quantum interference, and the reason a second Hadamard can turn an invisible phase into a visible, measurable flip.
Gates become rotations of this arrow: X is a half-turn about the x-axis that exchanges the poles (a bit flip), Z is a half-turn about the z-axis that leaves the poles fixed but rotates the equator by π (a phase flip), and H rotates the north pole onto the equator.
Once a gate is pictured as a rotation of an arrow on a sphere rather than an abstract matrix multiplication, single-qubit circuits stop feeling like symbol manipulation and start feeling geometric and, with practice, intuitive.
One limitation is worth flagging early: this is a picture of a single qubit's pure state only. It does not extend to two or more entangled qubits, and it needs a genuine modification — the density matrix, a few topics ahead — before it can represent a mixed state, where the corresponding point moves inside the sphere rather than sitting on its surface.
Try It Yourself
A qubit sits at Bloch angles θ = 60°, φ = 90°. Write out its state in α|0⟩+β|1⟩ form, and find the probability of measuring 0.
- 1Use the Bloch-vector form: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩, with θ = 60° so θ/2 = 30°.
- 2cos(30°) = √3/2, and sin(30°) = 1/2, so the amplitudes are α = √3/2 and β = e^{iπ/2}·(1/2) = i/2 (using φ = 90° = π/2 radians, and e^{iπ/2} = i from the Euler's-formula derivation in Complex Numbers & Dirac Notation).
- 3So |ψ⟩ = (√3/2)|0⟩ + (i/2)|1⟩.
- 4P(0) = |α|² = |√3/2|² = 3/4. (Check: P(1) = |i/2|² = 1/4, and 3/4+1/4 = 1. ✓)
|ψ⟩ = (√3/2)|0⟩ + (i/2)|1⟩, and P(measure 0) = 3/4 — note the phase φ affected β but never entered P(0), exactly as the geometry predicts.
Reference
| Bloch vector form | θ = polar angle, φ = azimuthal (phase) angle | |
| |0⟩ | North pole | |
| |1⟩ | South pole | |
| |+⟩ = (|0⟩+|1⟩)/√2 | Equator, reference phase | |
| |−⟩ = (|0⟩−|1⟩)/√2 | Equator, opposite phase |
Quick Check
Why can a single pure qubit state be fully described by just two real numbers (θ, φ)?
What does the Hadamard gate do to a Bloch vector starting at the north pole?