= Pauli multiplication identity
{c}
{title2=$\sigma_j\sigma_k=\delta_{jk}I+i\sum_\ell\epsilon_{jk\ell}\sigma_\ell$}
= Pauli matrix multiplication identity
{c}
{synonym}
Direct multiplication of the three <Pauli matrices> gives $X^2=Y^2=Z^2=I$, $XY=iZ$ and their cyclic permutations, with reversed products changing sign. Consequently $(\mathbf a\cdot\boldsymbol\sigma)(\mathbf b\cdot\boldsymbol\sigma)=(\mathbf a\cdot\mathbf b)I+i(\mathbf a\times\mathbf b)\cdot\boldsymbol\sigma$. In particular $(\mathbf s\cdot\boldsymbol\sigma)^2=|\mathbf s|^2I$, which gives the <eigenvalues> $(1\pm|\mathbf s|)/2$ of a qubit <density matrix>.
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