The Bloch sphere
The Bloch sphere is a way to picture any state of a single qubit as a point on a ball. The north pole is |0⟩, the south pole is |1⟩, the latitude sets the measurement probabilities, the longitude is the phase, and every single-qubit gate is a rotation of the sphere.
From two complex numbers to one point
In the previous lesson a qubit was two complex amplitudes, α|0⟩ + β|1⟩. That sounds like four real numbers, but two of them are not free:
- the probabilities must add up to one, |α|² + |β|² = 1;
- multiplying the whole state by a common phase (a global phase) changes nothing you could ever measure.
That leaves exactly two angles, and any state can be written as
Two angles are exactly what you need to name a point on the surface of a sphere — like latitude and longitude on a globe. That sphere is the Bloch sphere.
What θ and φ mean
- θ (theta) is measured down from the north pole, from 0° to 180°. It sets the balance between |0⟩ and |1⟩: θ = 0° is |0⟩, θ = 180° is |1⟩, θ = 90° (the equator) is an even 50/50 split.
- φ (phi) is the angle around the vertical axis, from 0° to 360°. It is the relative phase between the two amplitudes. It does not change the odds of measuring 0 or 1, but it changes how the qubit responds to later gates.
Drag the sliders and watch the arrow, the amplitudes and the probabilities change together:
Reading probabilities from the picture
The height of the arrow tip tells you the odds directly. If the tip is at height z (from −1 to +1):
So at the north pole (z = 1) you always measure 0, on the equator (z = 0) it is a coin flip, and the closer the arrow points down, the more likely a 1 becomes. In the explorer, the dashed cyan ring marks the arrow’s latitude.
The six states worth remembering
The points where the three axes cross the sphere have their own names:
| State | Formula | Where | (x, y, z) |
|---|---|---|---|
| |0⟩ | |0⟩ | north pole | (0, 0, 1) |
| |1⟩ | |1⟩ | south pole | (0, 0, −1) |
| |+⟩ | (|0⟩ + |1⟩)/√2 | equator, +X | (1, 0, 0) |
| |−⟩ | (|0⟩ − |1⟩)/√2 | equator, −X | (−1, 0, 0) |
| |+i⟩ | (|0⟩ + i|1⟩)/√2 | equator, +Y | (0, 1, 0) |
| |−i⟩ | (|0⟩ − i|1⟩)/√2 | equator, −Y | (0, −1, 0) |
The general conversion from angles to coordinates is the usual one for a sphere:
Gates are rotations
The real power of the picture: every operation you can do on one qubit is a rotation of the sphere around some axis. The X gate turns it 180° around the X axis, Z turns it 180° around the vertical axis, and the Hadamard gate turns it 180° around the diagonal between X and Z. While a gate runs below, its rotation axis lights up in cyan and the tip leaves a trail.
The arrow points at the north pole, so measuring now gives 0 every time. Press a gate to rotate it, or drag the sphere to look around.
Notice that Z, S and T never change P(0) and P(1): they spin the arrow around the vertical axis, so its height stays the same and only the phase changes. The quantum gates lesson goes through each gate in detail.
What the Bloch sphere cannot show
The sphere describes exactly one qubit. Two qubits need four complex amplitudes, and once they are entangled no pair of separate arrows can describe them — each individual qubit’s arrow shrinks into the inside of its sphere. Points inside the sphere are called mixed states. We will meet them again in the entanglement lesson.
Summary
- Every pure state of one qubit is a point on the surface of the Bloch sphere.
- North pole = |0⟩, south pole = |1⟩, equator = 50/50 superpositions.
- Latitude (θ) sets the probabilities: P(0) = (1 + z)/2. Longitude (φ) is the relative phase.
- Single-qubit gates are rotations of the sphere.
Frequently asked questions
Why are |0⟩ and |1⟩ opposite each other on the Bloch sphere instead of at right angles?
In the state formula the angle θ appears as θ/2. States that can be told apart perfectly (orthogonal states, like |0⟩ and |1⟩) are 90° apart in the underlying vector space, which becomes 180° — opposite points — on the sphere. That is why |+⟩ and |−⟩ are also opposite each other.
What is inside the Bloch sphere?
Points on the surface are pure states. Points inside the sphere (the “Bloch ball”) are mixed states: statistical mixtures, or a qubit that is entangled with something else. The centre is the completely mixed state — a fair coin with no phase information at all.
Can the Bloch sphere show two or more qubits?
Only partly. You can draw one sphere per qubit as long as the qubits are not entangled. For entangled qubits the individual arrows shrink towards the centre, because the information lives in the correlations between qubits rather than in any single one.
Who is the Bloch sphere named after?
Felix Bloch, the Swiss-American physicist who shared the 1952 Nobel Prize in Physics for nuclear magnetic resonance. The same picture was used for spin and for polarisation of light (where it is called the Poincaré sphere) long before quantum computing.