Qubits, Bloch Sphere, and Measurement

Summary

This note gives a geometric and operational picture of single-qubit states and measurement, which you’ll use everywhere: gates, circuits, noise models, and error correction.

Notes

Single-qubit state and Bloch sphere

Any pure qubit state can be written (up to global phase) as:

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

with 0 ≤ θ ≤ π, 0 ≤ φ < 2π.

This corresponds to a point on the Bloch sphere with coordinates:
- x = sin θ cos φ
- y = sin θ sin φ
- z = cos θ

Useful named states:

|0⟩: north pole
|1⟩: south pole
|+⟩ = (|0⟩ + |1⟩)/√2: +x direction
|−⟩ = (|0⟩ − |1⟩)/√2: −x direction

Measurement in different bases

Computational (Z) basis: |0⟩, |1⟩
- Outcome probabilities given by |α|^2, |β|^2 if |ψ⟩ = α|0⟩ + β|1⟩.
X basis: |+⟩, |−⟩
Y basis: eigenstates of Pauli Y (phase-shifted superpositions).

Implementation idea: to measure in a different basis, you often rotate the state then measure in Z.

Multi-qubit states and measurement

For n qubits, basis states are |x⟩ for x ∈ {0,1}^n; measurement gives:

a bitstring outcome x with probability |α_x|^2
post-measurement state |x⟩ (in that basis)

You can also measure subsets of qubits, leaving the rest in a (generally mixed) conditional state.

Projective measurement vs POVMs (why you care)

Most circuit-level descriptions use projective measurements in Pauli bases.
More general measurements (POVMs) appear in:
- realistic hardware models
- optimization of readout schemes
- security proofs in QKD

For this note, you can treat measurements as projectors plus classical outcomes.

Noise and decoherence (brief)

Noisy processes move states toward the center of the Bloch sphere (loss of purity) and/or rotate them randomly.

Common single-qubit noise channels:

Depolarizing: random Pauli with small probability.
Dephasing: shrinks coherence between |0⟩ and |1⟩ (Z-axis “squash”).
Amplitude damping: relaxation from |1⟩ toward |0⟩.

These are the building blocks for error models used in benchmarking and error correction.

References

03-quantum-mechanics-primer.md