InteractiveQuantum information

A Qubit, Many Possibilities

Connect a single qubit’s equation, geometry, and measurement probabilities.

Reference checked · 2 sources · View evidence

Field

Physics

Topic

Quantum Information

Estimated Time

12 min

Choose a perspective

Read, connect, and try the idea. Different formats offer ways into the same concept.

Definition

A qubit has two basis states, |0⟩ and |1⟩. A normalized pure state can be written as |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩. The squared amplitude magnitudes give measurement probabilities in this basis. This lesson describes one pure qubit, with global phase omitted.

Optional browser narration. Voices may run on your device or use a browser service; offline availability varies. Symbols may be pronounced imperfectly.

Explore the model

A qubit on the Bloch sphere

A 2D SVG projection of a 3D mathematical sphere for one pure qubit. It shows neither a particle’s position nor noise, mixed states, or entanglement. Global phase is omitted.

Bloch sphere state at theta 90 degrees, phi 0 degreesProjected state vector: x 1.000, y 0.000, z 0.000. Measuring in the zero, one basis gives zero with 50.0 percent probability and one with 50.0 percent probability.xyz · |0⟩|1⟩
Fixed viewing angle; the dot’s projected position can overlap. Coordinates below describe the full state.

0° selects |0⟩; 180° selects |1⟩. Use the arrow keys to adjust.

Changes the azimuth around z. It does not change these measurement probabilities. At either pole, phase makes no observable difference.

Try θ = 90°. Before moving φ, predict which numbers will change.

The same state, in symbols

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩

|ψ⟩ ≈ 0.707|0⟩ + (0.707 + 0.000i)|1⟩

i is the imaginary unit; numeric amplitudes are rounded. Bloch coordinates: x 1.000, y 0.000, z 0.000.

P(0) = cos²(θ/2) = 50.0%

P(1) = sin²(θ/2) = 50.0%

These are ideal probabilities across repeated preparations measured in the |0⟩, |1⟩ basis, not a prediction of a single measurement.

What you should take away

  • Read the two amplitudes in a single-qubit state.
  • Use theta and phi to explore a Bloch sphere.
  • Distinguish measurement probability from relative phase.
  • Explain why this model does not represent entanglement.

Pause and retrieve

Quick check

Keep θ at 90° and change φ from 0° to 90°. What happens to measurement probabilities in the |0⟩, |1⟩ basis?

Choose an answer to see the explanation. A wrong turn is part of learning.

Sources & evidence

These selected references support the explanation. They are not a live literature search or an exhaustive review.

Reference checked · 2026-10-06 · Educational sample

Educational sample: references and basic claims checked against the linked sources. This is not a claim of human expert editorial approval.

  1. Reference

    Superposition with Qiskit · The qubit state as a Bloch vector

    IBM Quantum Education · IBM Quantum Learning

    Supports: Pure single-qubit state equation, amplitudes, and relative phase.

    Scope: A teaching model, not a simulation of quantum hardware.

    Checked 2026-10-06

  2. Reference

    General Formulation of Quantum Information · Bloch Sphere

    John Watrous · IBM Quantum Learning

    Supports: Bloch sphere coordinates and the distinction between pure and mixed states.

    Checked 2026-10-06

Explore further

Use these discovery tools to find more perspectives. Search results need checking before they can support a claim.

Search Google Scholar (opens in a new tab)Explore Wikipedia (opens in a new tab)