Bloch Sphere

The one-qubit map

A single qubit can be any superposition of two basis states, written |0⟩ and |1⟩. The general pure state is

|ψ⟩ = α|0⟩ + β|1⟩,   with |α|² + |β|² = 1

Naively that's two complex numbers — four real parameters. But two of them don't matter for physics: the normalization constraint removes one, and the global phase (multiplying the whole state by e^iγ) is physically invisible, removing another. What's left is exactly two real numbers, and two real parameters are precisely the latitude and longitude of a point on a sphere.

That sphere is the Bloch sphere. Writing the two free parameters as a polar angle θ and an azimuthal angle φ:

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩

θ ∈ [0, π]   (polar angle from +z, the |0⟩ axis)
φ ∈ [0, 2π)  (azimuth around z — the relative phase)

The point on the unit sphere is then (x, y, z) = (sin θ cos φ, sin θ sin φ, cos θ). North pole is |0⟩, south pole is |1⟩, and everything in between is a genuine superposition. This single picture is the workhorse of every quantum-computing course and every qubit-control lab.

Poles, equator, and the factor of two

The most surprising feature is the θ/2 inside the cosine and sine. States that are orthogonal in Hilbert space — like |0⟩ and |1⟩ — end up on opposite ends of the sphere, 180° apart, even though their state vectors are 90° apart in the abstract complex space. The half-angle is the bridge between the abstract geometry and the everyday geometry of a globe.

State	(θ, φ)	Bloch vector (x, y, z)	Location
|0⟩	(0, —)	(0, 0, +1)	North pole
|1⟩	(π, —)	(0, 0, −1)	South pole
|+⟩ = (|0⟩+|1⟩)/√2	(π/2, 0)	(+1, 0, 0)	Equator, +x
|−⟩ = (|0⟩−|1⟩)/√2	(π/2, π)	(−1, 0, 0)	Equator, −x
|i⟩ = (|0⟩+i|1⟩)/√2	(π/2, π/2)	(0, +1, 0)	Equator, +y
|−i⟩ = (|0⟩−i|1⟩)/√2	(π/2, 3π/2)	(0, −1, 0)	Equator, −y

The equator is where superposition is most balanced: measuring in the computational basis gives 0 or 1 with equal 50/50 probability. The only difference between the four equatorial states above is their relative phase φ — invisible to a computational-basis measurement, but decisive once you rotate the basis.

The math: density matrix and Pauli vector

The cleanest way to define the Bloch sphere is through the density matrix ρ, which works for both pure and mixed states. Any single-qubit density matrix can be written using the three Pauli matrices σ = (σ_x, σ_y, σ_z):

ρ = ½ (I + r · σ) = ½ (I + x σ_x + y σ_y + z σ_z)

σ_x = [[0,1],[1,0]]   σ_y = [[0,−i],[i,0]]   σ_z = [[1,0],[0,−1]]

The real vector r = (x, y, z) is the Bloch vector. Its components are just the expectation values of the Pauli operators:

x = ⟨σ_x⟩,   y = ⟨σ_y⟩,   z = ⟨σ_z⟩
|r| = √(x² + y² + z²)

The length of the Bloch vector tells you the purity:

• |r| = 1 → a pure state, on the surface. Purity Tr(ρ²) = 1.
• 0 ≤ |r| < 1 → a mixed state, inside the ball. Tr(ρ²) = (1 + |r|²)/2.
• |r| = 0 → the maximally mixed state ρ = I/2, the exact center — a 50/50 classical coin with zero coherence.

So the full object is a solid ball; the "sphere" is just its pure-state surface. Noise and decoherence drag r inward, which is why the Bloch ball is the standard canvas for drawing the action of noise channels.

Gates as rotations

A single-qubit gate is a 2×2 unitary U. Because U preserves inner products and norms, it maps the unit sphere rigidly onto itself — and rigid maps of a sphere are exactly rotations. Every single-qubit gate (up to an irrelevant global phase) is a rotation of the Bloch vector about some axis n by some angle α:

R_n(α) = exp(−i (α/2) n·σ) = cos(α/2) I − i sin(α/2)(n·σ)

The half-angle appears again: a 2π rotation takes the underlying spinor to minus itself, so only a 4π rotation returns it — but the Bloch vector, living in SO(3), comes back after a full 360°. Here are the gates you'll actually meet:

Gate	Rotation axis	Angle	Effect on a Bloch point
Pauli-X (NOT)	x	180°	Flips north ↔ south (|0⟩ ↔ |1⟩)
Pauli-Y	y	180°	|0⟩ → i|1⟩, swaps poles with a phase
Pauli-Z	z	180°	Leaves poles fixed; flips equator phase
Hadamard (H)	(x+z)/√2	180°	|0⟩ → |+⟩: pole to equator
Phase gate S	z	90°	Spins the point around the equator
T gate (π/8)	z	45°	Small phase spin; key for universality
Rx(α), Rz(α)	x, z	α (any)	Continuous control rotations (lab pulses)

This is more than a metaphor. In a real lab a microwave or laser pulse is a rotation: its frequency sets the axis, its duration and amplitude set the angle. Calibrating a qubit gate literally means measuring how far a given pulse rotates the Bloch vector.

Measurement: the projection to a pole

Reading out a qubit in the computational basis asks a yes/no question: "0 or 1?" The probabilities come straight from the vector's height z = cos θ:

P(0) = cos²(θ/2) = (1 + z)/2
P(1) = sin²(θ/2) = (1 − z)/2

A point on the equator (z = 0) gives a perfect 50/50; a point near the north pole almost always returns 0. After the measurement, the state collapses: the Bloch vector jumps discontinuously to the north pole (if you got 0) or the south pole (if you got 1). Everything about φ and the in-plane direction is destroyed. That irreversible snap to a pole is the geometric face of wavefunction collapse — and the reason a single qubit, despite holding a continuum of (θ, φ), still yields only one classical bit.

Worked example: a Hadamard, then a phase, then read out

Start at the north pole, |0⟩, at (θ, φ) = (0, —), Bloch vector (0, 0, 1).

1. Apply H. Rotate 180° about the (x+z)/√2 axis. The north pole swings down to the +x point of the equator: state |+⟩, vector (1, 0, 0), (θ, φ) = (π/2, 0). Measuring now would give 0 or 1 with probability cos²(45°) = 0.5 each.
2. Apply S (90° about z). The point rotates a quarter-turn around the equator from +x to +y: state |i⟩, vector (0, 1, 0), (θ, φ) = (π/2, π/2). The measurement probabilities are unchanged — still 50/50 — because z didn't move. Only the phase φ advanced.
3. Apply H again. This maps the +y equatorial point to... still the equator (H rotates the (x,z) frame, leaving y mapped to −y). To make the phase visible, instead apply H to the |+⟩ state from step 1: it returns the point to the north pole, |0⟩, and now a measurement gives 0 with probability 1.

The lesson: the phase φ added by S is invisible to a direct measurement, but a second Hadamard converts that phase into a measurable population difference. Turning hidden phase into measurable amplitude through interference is the engine of every quantum algorithm — from Deutsch–Jozsa to Grover to Shor.

Where the Bloch sphere shows up

• Quantum computing hardware. Superconducting transmons, trapped ions, and neutral atoms are all controlled as Bloch-sphere rotations. A typical single-qubit gate on IBM/Google superconducting hardware runs in ~20–40 ns with fidelities above 99.9%; the calibration loop measures rotation axes and angles directly.
• NMR and MRI. The Bloch sphere descends from Felix Bloch's 1946 equations for nuclear spins precessing in a magnetic field. The "Bloch vector" precessing about the field is literally a magnetic moment; MRI contrast comes from how fast different tissues' vectors relax back along the field (T₁) and dephase in-plane (T₂).
• Qubit calibration (Rabi & Ramsey). A Rabi experiment drives continuous rotation and reads off the rotation rate; a Ramsey experiment puts the vector on the equator and watches the phase precess, measuring the dephasing time T₂* directly as the Bloch vector spirals inward.
• Quantum optics. The polarization of a single photon is a qubit; its Poincaré sphere is mathematically identical to the Bloch sphere, with horizontal/vertical polarization at the poles and diagonal/circular states on the equator.
• Quantum error correction. Noise channels (bit-flip, phase-flip, depolarizing, amplitude damping) each have a clean geometric action — shrinking, squashing, or biasing the Bloch ball — which is how their effect on logical qubits is reasoned about.

Common misconceptions and edge cases

• "The sphere holds infinite information." A qubit's (θ, φ) is a continuum, but you can never read it out: one measurement yields one bit, and the state collapses. Holevo's theorem caps the accessible classical information at exactly one bit per qubit.
• "Two qubits = two spheres." Only for product (unentangled) states. Entangled states have no separate-sphere description at all; each qubit's local state sits at the center of its own ball (maximally mixed), with all the information hiding in the correlations.
• "The factor of 2 is a convention." It's forced: it's the 2-to-1 covering map from SU(2) (state vectors / spinors) onto SO(3) (Bloch-vector rotations). The same factor is why a spin-½ needs a 720° turn to return to itself.
• "Global phase is on the sphere." It isn't — global phase is quotiented out, which is the whole reason 4 real parameters collapse to 2. Only the relative phase φ is geometric.
• "Inside the ball is forbidden." The interior is perfectly physical — it's the mixed states. A pure state hit by decoherence moves inward; that's not an error, it's the geometry of losing quantum coherence.
• "Measurement probabilities depend on φ." In the computational basis they depend only on θ (the height z). φ matters only after you rotate the measurement basis — that's interference, not direct readout.