= Pauli operator basis
{c}
{title2=$\{I,X,Y,Z\}$}
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> $\operatorname{Tr}(P^\dagger Q)=2\delta_{PQ}$. Thus every operator has the expansion $A=\frac12\sum_{P\in\{I,X,Y,Z\}}\operatorname{Tr}(PA)P$. 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.
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