 Pauli operator basis

The identity and the three Pauli matrices form an orthogonal basis for the complex vector space of two-by-two matrices, with Hilbert-Schmidt inner product $\mathrm{Tr}(P^{†}Q)=2{\delta }_{PQ}$. Thus every operator has the expansion $A=\frac{1}{2}{\sum }_{P\in \{I,X,Y,Z\}}\mathrm{Tr}(PA)P$. For a density matrix, the identity coefficient is $1/2$ and the remaining real coefficients are half the components of its Bloch vector. This converts unitary conjugation into transformations of three real components.