Haar twirling conditional expectation (${\mathcal{E}}_A(O)=d_B^{-1}{\mathrm{Tr}}_B(O)\otimes I_B$)

For an operator on a tensor-product Hilbert space, averaging unitary conjugations on subsystem $B$ with normalized Haar measure gives ${\mathcal{E}}_A(O)=({\mathrm{Tr}}_{B}O/d_B)\otimes I_B$. It fixes precisely the operators acting trivially on $B$ and is contractive in operator norm. Writing $O-UO{U}^{†}=[O,U]U^{†}$ converts localization error into a commutator bound.