negligible in $n$ such that for all adversaries $\tilde{\mathcal{A}}$ and for all steps $i$ we have: $$\Delta\qty(v_{i}(\tilde{\mathcal{A}}, \rho_{in}), \Tr_{\mathcal{B}_{i}}\qty(\tilde{\rho}_{i}(\tilde{A}, \rho_{in}))) \leq \epsilon(n)$$ where $v_{i}(\tilde{\mathcal{A}}, \rho_{in})$ is the view of the adversary when interacting with the simulator and $\Tr_{\mathcal{B}_{i}}\qty(\tilde{\rho}_{i}(\tilde{A}, \rho_{in}))$ is the view of the adversary in the real protocol.

For further details on these definitions, see [@DNS10]. Another property which a quantum protocol may satisfy is *verifiability*. This intuitively means that the probability of receiving a corrupted output without aborting is negligible.

\[verification\] A protocol is said to be *$\epsilon$-verifiable for party $P_{i}$* if for any (potentially malicious) behaviour of party $P_{j}$ with $j \neq i$, the probability of obtaining a wrong output and not aborting is bounded by $\epsilon$. If the output of the real protocol with malicious party $\tilde{P}_{j}$ is $\tilde{\rho}(\tilde{P}_{j}, \rho_{in})$ then we have that: where

[$$\begin{array}[c]{lllllllllllllll}\rho_{ideal}(\rho_{in})&:=&p_{ok}(\bbbone_{\H_{P_i}}\otimes \mathcal{C}_{\H_{P_j}})\cdot U_f \cdot(\rho_{in})+(1-p_{ok})([\mathsf{abort}_1])\end{array}$$]{} where $\mathcal{C}_{\mathcal{H}_{P_{j}}}$ is the deviation that acts on $P_{j}$’s systems after they receive their outcome (a CP-map, it can be purified by including ancilla), $\rho'_{in} = (\bbbone_{\mathcal{H}_{P_i}} \otimes \mathcal{D}_{\mathcal{H}_{P_{j}}})\rho_{in}$ is an initial state compatible with $P_{i}$’s input, where $\mathcal{D}_{\mathcal{H}_{P_{j}}}$ is a deviation on the input by $P_{j}$.

Note that since $\mathcal{C}_{\H_{P_j}}$ is performed at the final step 