Quantum verifier acceptance operator ($M=(I\otimes ⟨0|)V^{†}{\Pi }_{\mathrm{acc}}V(I\otimes |0⟩)$)

A verifier with clean ancillas and a designated output measurement induces an effect $0\le M\le I$ on the witness register. Its acceptance probability is $\mathrm{Tr}(M\rho )$, and the largest possible acceptance is $\Vert M\Vert$. Spectral bounds on $M$ make the soundness quantifier over all witnesses, including entangled repeated witnesses, explicit.