Generalized Pauli matrix ($\mathrm{Tr}({\sigma }_k{\sigma }_{\ell })={\delta }_{k\ell }$)

A traceless Hermitian matrix in a Hilbert-Schmidt inner product orthonormal basis of the real vector space of traceless Hermitian $N$-by-$N$ matrices. One choice uses symmetric and imaginary antisymmetric off-diagonal matrix units, normalized by $1/\sqrt{2}$, and traceless diagonal generators. Together with $I/\sqrt{N}$ they form an orthonormal basis of all Hermitian matrices. The coordinates $s_k=\mathrm{Tr}(\rho {\sigma }_k)$ give the Generalized Bloch representation.