The Bloch sphere

One qubit as a point on a sphere: angles, rotations and what entanglement does to the picture.

You will learn to

Place |0⟩, |1⟩, |+⟩, |−⟩ on the sphere.
Read RX, RY, RZ as rotations about the three axes.
Explain 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.

∣ ψ ⟩ = cos \frac{θ}{2} ∣0 ⟩ + e^{i φ} sin \frac{θ}{2} ∣1 ⟩

Gates 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

Try it

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];

Open in the IDE Open in the Playground

Short 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.

Try it

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];

Open in the IDE Open in the Playground

Quiz 1 · 10 XP

Which gate moves a point along the equator without changing measurement probabilities in the Z basis?

Quiz 2 · 10 XP

A qubit's Bloch arrow has length 0. What does that mean?

Key takeaway

One pure qubit is a point on the Bloch sphere; single-qubit gates are rotations of that point. A short arrow means the qubit is entangled and has no state of its own.

Docs: Q-sphere and Bloch views

Up next: Pauli gates: X, Y, Z · Gates and Circuits

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

half turn about the X+Z diagonal

swaps the Z and X axes