Quantum verifier acceptance operator
= Quantum verifier acceptance operator
{title2=$M=(I\otimes\langle0|)V^\dagger\Pi_{\rm acc}V(I\otimes|0\rangle)$}
A verifier with clean ancillas and a designated output measurement induces an effect $0\leq M\leq I$ on the witness register. Its acceptance probability is $\operatorname{Tr}(M\rho)$, and the largest possible acceptance is $\|M\|$. Spectral bounds on $M$ make the soundness quantifier over all witnesses, including entangled repeated witnesses, explicit.