Bloch Sphere

Canonical treatment: Bloch Sphere maintains the unified pure-state, mixed-state, and rotation geometry with exercises and references.

This bridge is intentionally limited to the density-operator application: mixed-qubit parametrization, purity, and measurement predictions.

Bloch-vector dictionary

Every qubit state is uniquely

$$\rho=\frac12(I+\mathbf r\cdot\boldsymbol\sigma), \qquad \lVert\mathbf r\rVert\le1.$$

Positivity gives the unit-ball condition. Pure states satisfy $\lVert\mathbf r\rVert=1$, the maximally mixed state has $\mathbf r=0$, and

$$\operatorname{Tr}(\rho^2)=\frac12(1+\lVert\mathbf r\rVert^2).$$

For a Pauli measurement along unit vector $\mathbf n$,

$$p(\pm|\mathbf n)=\frac12(1\pm\mathbf r\cdot\mathbf n).$$

The boundary is the Bloch sphere; the full mixed-state space is the Bloch ball. Continue to the canonical page for spinors, relative phase, and the $SU(2)$–$SO(3)$ rotation map.