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The Bloch sphere
One qubit as a point on a sphere: angles, rotations and what entanglement does to the picture.
- 01Place |0⟩, |1⟩, |+⟩, |−⟩ on the sphere.
- 02Read RX, RY, RZ as rotations about the three axes.
- 03Explain why an entangled qubit's arrow is short.
Up to an overall phase, any single-qubit state is fixed by two angles: how far from |0⟩ (θ) and which way round (φ). Plot those angles on a sphere and every pure state is a point on its surface. |0⟩ is the north pole, |1⟩ the south pole, |+⟩ and |−⟩ sit on the equator.
01Gates are rotations
| Gate | Rotation | Example |
|---|---|---|
| rx(θ) | angle θ about the X axis | rx(pi) = X (up to phase) |
| ry(θ) | angle θ about the Y axis | ry(pi/2)|0⟩ = |+⟩ |
| rz(θ) | angle θ about the Z axis: changes phase only | rz(pi/2) ≈ S, rz(pi/4) ≈ T |
| h | half turn about the X+Z diagonal | swaps the Z and X axes |
Drag an angle
Select the RY gate in the composer and drag its angle slider while the State tab shows the Bloch view. The arrow sweeps from the north pole down through the equator.
OPENQASM 3.0;
include "stdgates.inc";
qubit[1] q;
ry(pi/3) q[0];
02Short arrows: entanglement
The Bloch sphere describes one qubit on its own. When a qubit is entangled with another, its own state is mixed: no single point on the surface describes it, and QubitLab draws a shorter arrow inside the sphere. The length is the purity. A maximally entangled qubit sits at the centre: every measurement on it alone is a coin flip.
Watch the arrow shrink
State tab → Bloch. After H the first qubit has a full-length arrow on the equator; after CX both arrows collapse to the centre and are tagged 'entangled'.
OPENQASM 3.0;
include "stdgates.inc";
qubit[2] q;
h q[0];
cx q[0], q[1];
2 questions · one try each counts
Which gate moves a point along the equator without changing measurement probabilities in the Z basis?
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A qubit's Bloch arrow has length 0. What does that mean?
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