Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 35B / c / Solution

For a mixed spin-half state, the Bloch vector formula givesPhysical consistency is equivalent to the resulting Bloch vector having length at most one.

Three linearly independent Hermitian operators do not always suffice. The operatorsare linearly independent, but the first expectation is always one and none detects the $y$ component. States with Bloch vectors $(0,c,0)$ and $(0,-c,0)$ have identical expectations for all three. The correct criterion for informationally complete three-observable qubit tomography is that the three traceless parts span the Pauli space.