consequences for the parties, eg. financial repercussions. By associating the correct cost to being caught, even if the probability of getting caught is not exponentially close to $1$ the deterrence might be still high enough to make cheating unappealing.

We say that the protocol is $\epsilon$-private for $B$ if for all adversaries $\tilde{\mathcal{A}}$ and for all steps $i$ we have: $$\Delta\qty(v_{i}(\tilde{\mathcal{A}}, \rho_{in}), \Tr_{\mathcal{B}_{i}}\qty(\tilde{\rho}_{i}(\tilde{A}, \rho_{in}))) \leq \epsilon$$

Note that we do not have any requirement on $\epsilon$ and so, while we would like it to be close to $0$ it does not necessarily need to be negligible.

Stronger and more elegant definitions of covert adversaries that might be adaptable to the quantum case can be found in [@AumLin07], but for reasons specific to our construction (namely the fact that measurement is irreversible and disturbs quantum states) they are not directly applicable here.

#### Relation with specious and malicious adversaries

In the classical case the covert adversary lies between the honest-but-curious and malicious ones for some choices of $\epsilon$, as shown in [@AumLin07]. Their analysis partially holds also for the quantum case: the quantum covert adversary is strictly less powerful than the fully malicious adversary, unless the $\epsilon$ is negligible, in which 