Heralded success and conditional verification contract

A heralded procedure has an explicit finite outcome set containing success and failure, a success probability $p_s$, and a declared output type. For an input density operator $\rho$ as in Density operator, its success branch is the subnormalized state ${\mathcal{I}}_s(\rho )$ of the instrument Finite instruments and conditional states, with $p_s=\mathrm{tr}({\mathcal{I}}_s(\rho ))$. If $p_s>0$, its conditional output state is ${\rho }_s={\mathcal{I}}_s(\rho )\mathrm{/}{p}_s$; if $p_s=0$, no normalized conditional state is assigned and the branch is denoted $\mathrm{\perp }$. The outcome law is specified by the instrument and the POVM conventions of Finite POVMs and their outcome law.

The contract states separately (i) the success probability, (ii) the guarantee conditional on success, (iii) any unconditional guarantee including failure, and (iv) the cost of verification and of outputting or recording the success flag. An acceptance probability from Quantum acceptance probability for a finite promise computation is not itself a conditional-state guarantee. A claim about a quantum state output uses the type and cost distinctions in Classical output, quantum-state output, and access costs.