Displaced-state concentration: the exact criterion

For a self-adjoint operator $A$ in a state $\rho$, its spectral law is ${\mu }_{\rho ,A}(J)=\mathrm{Tr}(\rho E_A(J))$. For probability laws with finite second moments, $W_2$ denotes the quadratic Wasserstein metric. In particular,

because every coupling with a point mass has constant second coordinate.