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Bloch Sphere

The Bloch sphere is a geometrical representation of the pure state space of a single qubit, mapping every possible qubit state to a point on the surface of a unit sphere.

What it means

Any pure single-qubit state can be written as |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩, where θ (polar angle) and φ (azimuthal angle) define a point on the Bloch sphere.The north pole corresponds to |0⟩, the south pole to |1⟩, and the equator contains equal superpositions with various relative phases.Single-qubit gates correspond to rotations of this sphere.The Bloch sphere provides crucial geometric intuition: the X, Y, Z Pauli gates are 180-degree rotations around the x, y, z axes respectively, and the Hadamard gate maps between the z and x axes.Mixed states (from decoherence) map to points inside the sphere, not on its surface.

Everyday analogy

The Bloch sphere is like a globe where the north pole is |0⟩ and the south pole is |1⟩ — every quantum state has its own 'latitude and longitude' on this globe, and quantum gates are like rotating the globe.
Think of it as a compass for quantum states — just as a compass needle can point in any direction on a sphere, a qubit's state can correspond to any point on the Bloch sphere.
Where does the name come from? Not from any property of the sphere — it honors the Swiss-American physicist Felix Bloch (Nobel Prize 1952, nuclear magnetic resonance). A person's name simply became the term.

Common misconceptions

  • The Bloch sphere only works for SINGLE qubits — there is no simple geometric representation for multi-qubit states, which live in exponentially larger Hilbert spaces.
  • Points INSIDE the Bloch sphere are not invalid — they represent mixed states (statistical mixtures), while points on the surface represent pure states.
  • The Bloch sphere is NOT a literal shape — a qubit is not a ball and has no spatial direction; the sphere is only a visualization of two complex amplitudes. Treating it as physical geometry is a common source of confusion.

Key takeaways

  • Every pure single-qubit state maps to a unique point on the surface of the Bloch sphere, parameterized by angles θ and φ.
  • Quantum gates on a single qubit correspond to rotations of the Bloch sphere.
  • The Bloch sphere provides geometric intuition for understanding single-qubit operations and states.
  • The Hadamard gate is a 90° rotation about the Y axis followed by a 180° rotation about the X axis — that second rotation is what makes H self-inverse (H·H = I) for every initial state.

Check your understanding

On the Bloch sphere, which state is located at the north pole?

  1. A.|1⟩
  2. B.|+⟩
  3. C.|0⟩
  4. D.|−⟩
Show the answer

Answer: C. |0⟩

Why: The north pole of the Bloch sphere corresponds to the state |0⟩ (θ=0), while the south pole corresponds to |1⟩ (θ=π).

Builds on

QubitDirac (Bra-Ket) Notation

Primary source: F. Bloch, Nuclear Induction, Phys. Rev. 70, 460 (1946), doi:10.1103/PhysRev.70.460

Bloch-sphere representation standard per N&C (2010).

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