The Bloch Sphere — A Map of All States
You know qubits hold superpositions, collapse on measurement, and cannot be peeked at. Now the question: is there a map of all possible states? Yes — and it is a sphere.
How Many Distinct States Can a Qubit Be In?
A qubit can be in |0⟩, in |1⟩, or in any superposition between them. How many genuinely distinct pure states exist for a single qubit?
L05 introduced the superposition spectrum — the idea that infinitely many states exist between pure |0⟩ and pure |1⟩. But that picture was deliberately simplified: it showed only one dimension, and it missed something important.
In L05, changing the slider moved the qubit's probability — how likely it was to collapse to |0⟩ vs |1⟩. But L06 introduced a second property: phase. Two qubits can have identical measurement probabilities yet be completely different quantum states because their phases differ. Phase is invisible to a single measurement, but it is real — it governs interference, which is how quantum algorithms amplify the right answer.
A complete description of a qubit requires two independent angles: one for probability (θ) and one for phase (φ). Two angles means a surface. And the surface they trace out is a sphere.
The Quantum Globe
The Bloch sphere is easiest to understand through an analogy you already have: the Earth.
The Bloch sphere works exactly the same way. Every single-qubit state corresponds to one point on its surface, described by two angles: θ (the polar angle, like latitude) and φ (the azimuthal angle, like longitude).
The north pole is |0⟩ — measuring always gives 0. The south pole is |1⟩ — measuring always gives 1. The equator is perfect 50/50 superposition. Your qubit's location on this globe completely determines everything about it.
When you measure, the arrow snaps to the north pole or south pole.
θ, φ and the Six Axis States
θ — the polar angle — controls probability
θ runs from 0° at the north pole (|0⟩) to 180° at the south pole (|1⟩), passing through 90° at the equator. This angle directly determines what you will get when you measure:
P(measuring |0⟩) = cos²(θ/2) · P(measuring |1⟩) = sin²(θ/2)
The factor of ½ is not arbitrary — it ensures that antipodal points on the sphere (points directly opposite each other) correspond to orthogonal states. |0⟩ and |1⟩ are opposite poles, 180° apart on the sphere, and they are perfectly orthogonal. Without the ½, this geometric elegance would break.
φ — the azimuthal angle — controls phase
φ sweeps 0° to 360° around the equator. Unlike θ, changing φ does not change the measurement probabilities at all — two states at the same θ but different φ give identical statistics in the Z-basis. What φ encodes is relative phase — the phase relationship between the |0⟩ and |1⟩ components of the superposition. This phase is invisible to direct measurement but is precisely what quantum interference exploits.
The six axis states — the vocabulary of quantum computing
The three axes of the Bloch sphere define six special points that appear constantly in quantum circuits, algorithms, and measurement protocols.
| State | Name | Location | θ | φ | P(0) / P(1) |
|---|---|---|---|---|---|
| |0⟩ | Computational 0 | +Z — north pole | 0° | — | 100% / 0% |
| |1⟩ | Computational 1 | −Z — south pole | 180° | — | 0% / 100% |
| |+⟩ | Plus state | +X — equator | 90° | 0° | 50% / 50% |
| |−⟩ | Minus state | −X — equator | 90° | 180° | 50% / 50% |
| |i⟩ | Plus-i state | +Y — equator | 90° | 90° | 50% / 50% |
| |−i⟩ | Minus-i state | −Y — equator | 90° | 270° | 50% / 50% |
Notice that |+⟩ and |−⟩ have identical measurement probabilities — both give 50/50 — yet they sit at opposite ends of the sphere. They are completely different quantum states. The difference is entirely in phase (φ = 0° vs φ = 180°), which means they interfere constructively and destructively in opposite ways. This is the distinction the Hadamard gate exploits.
One Qubit Only — The Limits of the Sphere
Live 3D Bloch Sphere Explorer
Drag the sphere to orbit. Use sliders to move the state vector precisely. Click the six axis preset buttons to snap to key states. The dashed arcs label θ and φ live on the globe.
2. The six axis states: Click each preset button — |0⟩, |1⟩, |+⟩, |−⟩, |i⟩, |−i⟩. Notice |+⟩ and |−⟩ are both 50/50 but opposite on the sphere. Same measurement statistics, completely different states.
3. Orbit the sphere: Drag the globe to view from different angles. Notice the θ and φ arcs update live as you move the state with sliders.
What You Now Know About the Bloch Sphere
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Every point on the surface = one distinct pure qubit stateThe sphere is a complete, two-way map. Infinite states, each with its own precise character. North pole = |0⟩. South pole = |1⟩. Equator = perfect 50/50 superposition.
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θ controls probability, φ controls phaseP(|0⟩) = cos²(θ/2). The azimuthal angle φ encodes relative phase — invisible to Z-measurement but the engine of quantum interference. Two states at the same latitude but different longitude are completely different quantum states.
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Six axis states define the vocabulary|0⟩ (+Z), |1⟩ (−Z), |+⟩ (+X), |−⟩ (−X), |i⟩ (+Y), |−i⟩ (−Y). These six appear in every quantum circuit. |+⟩ and |−⟩ are both 50/50 but differ entirely in phase.
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One qubit only — entanglement breaks the sphere pictureThe Bloch sphere is perfect for one qubit. Two entangled qubits cannot be represented as two separate arrows on two spheres. The sphere ends where entanglement begins. Track 2 builds the right language for multi-qubit states.
One qubit lives on a sphere.
What happens when you have two?
The state space does not double — it squares.
- Nielsen, M. A. & Chuang, I. L. — Quantum Computation and Quantum Information, Cambridge, 2000. §1.2 — The Bloch sphere, state vectors, and qubit parameterisation.
- Bloch, F. — "Nuclear Induction," Physical Review, 70, 1946. — Original paper introducing the sphere representation for spin states.
- Preskill, J. — Ph219 Lecture Notes, Chapter 2. theory.caltech.edu/~preskill/ph219/ — State space geometry and qubit parameterisation.
- IBM Qiskit Textbook — "The Bloch Sphere": qiskit.org/learn — Practical introduction with circuit diagrams.