  Quantum resource theories provide a mathematically rigorous way of
understanding the nature of various quantum resources. An important problem in
any quantum resource theory is to determine how quantum states can be converted
into each other within the physical constraints of the theory. The standard
approach to this problem is to study approximate or probabilistic
transformations. Very few results have been presented on the intermediate
regime between probabilistic and approximate transformations. Here, we
investigate this intermediate regime, providing limits on both, the fidelity
and the probability of state transitions. We derive limitations on the
transformations, which are valid in all quantum resource theories, by providing
bounds on the maximal transformation fidelity for a given transformation
probability. As an application, we show that these bounds imply an upper bound
on the asymptotic rates for various classes of states under probabilistic
transformations. We also show that the deterministic version of the single copy
bounds can be applied for drawing limitations on the manipulation of quantum
channels, which goes beyond the previously known bounds of channel
manipulations. Furthermore, we completely solve the question of
stochastic-approximate state conversion via local operations and classical
communication in the following two cases: (i) Both initial and target states
are pure bipartite entangled states of arbitrary dimensions. (ii) The target
state is a two-qubit entangled state and the initial state is a pure bipartite
state.
