  Protocols for discriminating between a pair of channels or for estimating a
channel parameter can often be aided by adaptivity or by entanglement between
the probe states. For large numbers of channel uses, this can make it difficult
to bound the best possible performance for such protocols. In this paper, we
introduce a quantity that we call the relative fidelity of a given pair of
channels and a pair of input states to those channels. Constraining the allowed
input states to all pairs of states whose fidelity is greater than some minimum
"input fidelity" and minimising this quantity over the valid pairs of states,
we get the minimum relative fidelity for that input fidelity constraint. We are
then able to lower bound the fidelity between the possible output states of any
protocol acting on one of two possible channels in terms of the minimum
relative fidelity. This allows us to bound the performance of discrimination
and parameter estimation protocols, as well as to rule out perfect
discrimination for certain pairs of channels, even using the most general,
adaptive protocols. By finding a continuity bound for the relative fidelity, we
are also able to provide a simple proof that the quantum Fisher information
(QFI) of the output of an $N$-use protocol is no more than $N^2$ times the
one-shot QFI.
