Understanding qubits

Two amplitudes, one state

A pure qubit lives in a two-dimensional complex vector space. After removing an unobservable global phase, we can write it as

$$\lvert\psi\rangle=\cos(\theta/2)\lvert0\rangle+ e^{i\phi}\sin(\theta/2)\lvert1\rangle.$$

The polar angle $\theta$ sets the Z-basis probabilities. The azimuth $\phi$ sets the relative phase. The explorer uses degrees for its controls; the formula uses radians.

Follow the sphere

At $\theta=0$, the state is $\lvert0\rangle$. At $\theta=180°$, it is $\lvert1\rangle$, up to a global phase. Along the equator, $\theta=90°$ and both Z outcomes have probability $1/2$.

L01 · INTERACTIVE LABRuns on your device

Qubit explorer

A pure qubit is a normalised pair of complex amplitudes. The sphere shows the state; the bars show measurement probabilities in the Z basis.

Make a prediction first

Move the state to the equator. What will happen to the Z-basis probabilities when you change only the phase?

Experiment controls

Polar angle θ90°Relative phase φ0°

HXYZST

Prepare |0⟩Random seed

Run measurements →Reset

Exact probability

|0⟩

50.0%

|1⟩

50.0%

Exact ideal-model predictions · run a measurement experiment to compare.

MODEL · Ideal pure-state vector, complex double precisionORDER · q0 is the leftmost bit

Take it further

Create a state with a 75% probability of measuring 1. Can you find two different phases with the same probabilities?

Prediction: Move the state to the equator. What will happen to the Z-basis probabilities when you change only the phase? Challenge: Create a state with a 75% probability of measuring 1. Can you find two different phases with the same probabilities?

Work one example

Choose $\theta=120°$ and $\phi=90°$. Then

$$\lvert\psi\rangle=\frac12\lvert0\rangle+i\frac{\sqrt3}{2}\lvert1\rangle.$$

Therefore $P(0)=1/4$ and $P(1)=3/4$. A 1,024-shot experiment has expected counts 256 and 768, but its observed counts fluctuate.

What the picture does and does not mean

The arrow is a geometric representation of an ideal pure state. It is not a tiny physical arrow inside every device. Mixed states belong inside the Bloch sphere; this particular explorer stays on the surface and does not model decoherence.

PAUSE & THINK

You change only φ while keeping θ fixed. What stays the same?

Choose your answer

The entire quantum stateThe Z-basis measurement probabilitiesAll possible measurement results in every basis

Check my answer

Reference: Microsoft Learn — The qubit.