  Given two pairs of quantum states, we want to decide if there exists a
quantum channel that transforms one pair into the other. The theory of quantum
statistical comparison and quantum relative majorization provides necessary and
sufficient conditions for such a transformation to exist, but such conditions
are typically difficult to check in practice. Here, by building upon work by
Matsumoto, we relax the problem by allowing for small errors in one of the
transformations. In this way, a simple sufficient condition can be formulated
in terms of one-shot relative entropies of the two pairs. In the asymptotic
setting where we consider sequences of state pairs, under some mild convergence
conditions, this implies that the quantum relative entropy is the only relevant
quantity deciding when a pairwise state transformation is possible. More
precisely, if the relative entropy of the initial state pair is strictly larger
compared to the relative entropy of the target state pair, then a
transformation with exponentially vanishing error is possible. On the other
hand, if the relative entropy of the target state is strictly larger, then any
such transformation will have an error converging exponentially to one. As an
immediate consequence, we show that the rate at which pairs of states can be
transformed into each other is given by the ratio of their relative entropies.
We discuss applications to the resource theories of athermality and coherence.
