Represent a pure qubit state on the Bloch sphere by first removing its physically irrelevant global phase, then writing it as |ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩. Its point on the sphere is (x, y, z) = (sin θ cos φ, sin θ sin φ, cos θ), with 0 ≤ θ ≤ π and 0 ≤ φ < 2π. Pure states lie on the surface; mixed states occupy the interior of the Bloch ball.
Start with a normalized qubit state
A general pure qubit is |ψ⟩ = α|0⟩ + β|1⟩, where the complex amplitudes satisfy |α|² + |β|² = 1. Multiplying both amplitudes by the same phase factor does not change the physical state. When α is nonzero, choose that overall phase so α is real and nonnegative. The remaining phase difference between the two amplitudes is the relative phase φ.
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Normalization lets the two amplitude magnitudes be represented by a half-angle: |α| = cos(θ/2) and |β| = sin(θ/2). Thus the standard form is:
|ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩
Here θ ranges from 0 to π, and φ ranges from 0 (inclusive) to 2π (exclusive). The half-angle matters: the polar position on the sphere is θ, while the amplitudes use θ/2. This parameterization is the standard association between pure qubit states and points on the unit sphere, as described by IBM Quantum Learning.
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Convert the angles to Bloch coordinates
Use the ordinary spherical-coordinate convention: θ is measured down from the positive z-axis, and φ is measured around the z-axis from positive x toward positive y. The corresponding Cartesian coordinates are:
x = sin θ cos φy = sin θ sin φz = cos θ
In vector form, the point is (x, y, z) = (sin θ cos φ, sin θ sin φ, cos θ). Because x² + y² + z² = 1 for this pure-state form, the point lies on the unit sphere.
Read the same coordinates from the density matrix
The density matrix for the pure state is ρ = |ψ⟩⟨ψ|. It can be written in terms of the Pauli matrices as:
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Equivalently, ρ = 1/2(I + xX + yY + zZ). The coordinates x, y, and z are the expectation values of the Pauli observables X, Y, and Z, respectively. This makes the Bloch vector more than a geometric mnemonic: its components directly give the single-qubit Pauli expectation values.
Locate familiar states
The computational basis states are the poles; equal-amplitude states lie on the equator. These examples also check the sign convention for the y-coordinate:
Rank #4
| State | Angles or amplitudes | Bloch coordinates |
|---|---|---|
|0⟩ |
θ = 0 |
(0, 0, 1), north pole |
|1⟩ |
θ = π |
(0, 0, −1), south pole |
|+⟩ = (|0⟩ + |1⟩)/√2 |
θ = π/2, φ = 0 |
(1, 0, 0), positive x-axis |
|−⟩ = (|0⟩ − |1⟩)/√2 |
θ = π/2, φ = π |
(−1, 0, 0), negative x-axis |
|+i⟩ = (|0⟩ + i|1⟩)/√2 |
θ = π/2, φ = π/2 |
(0, 1, 0), positive y-axis |
|−i⟩ = (|0⟩ − i|1⟩)/√2 |
θ = π/2, φ = 3π/2 |
(0, −1, 0), negative y-axis |
Distinguish pure states from mixed states
A pure state has a rank-one density matrix and a unit-length Bloch vector, so it sits on the sphere’s surface. A general mixed state can be represented by a Bloch vector shorter than one, placing it inside the sphere. The maximally mixed state I/2 is the center, at (0, 0, 0). The full set of single-qubit density matrices therefore fills the Bloch ball, not just its surface; the sphere’s surface alone represents pure states.
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Account for the poles and multi-qubit limits
Azimuth is undefined at either pole
At |0⟩, θ = 0; at |1⟩, θ = π. At either point, changing φ does not change the state. This is a coordinate singularity of spherical angles, not a physical ambiguity in the pole state.
A local Bloch vector is not a full entangled state
For a multi-qubit system, a Bloch plot for one qubit shows that qubit’s individual X, Y, and Z expectation values. It does not encode correlations with the other qubits, so separate per-qubit plots cannot fully specify an entangled joint state. Treat such plots as local visualizations rather than complete representations of the entire system.
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