The quantum relative entropy is known to play a key role in determining the
asymptotic convertibility of quantum states in general resource-theoretic
settings, often constituting the unique monotone that is relevant in the
asymptotic regime. We show that this is no longer the case when one allows
stochastic protocols that may only succeed with some probability, in which case
the quantum relative entropy is insufficient to characterize the rates of
asymptotic state transformations, and a new entropic quantity based on a
regularization of Hilbert’s projective metric comes into play. Such a scenario
is motivated by a setting where the cost associated with transformations of
quantum states, typically taken to be the number of copies of a given state, is
instead identified with the size of the quantum memory needed to realize the
protocol. Our approach allows for constructing transformation protocols that
achieve strictly higher rates than those imposed by the relative entropy.
Focusing on the task of resource distillation, we give broadly applicable
strong converse bounds on the asymptotic rates of probabilistic distillation
protocols, and show them to be tight in relevant settings such as entanglement
distillation with non-entangling operations. This generalizes and extends
previously known limitations that only apply to deterministic protocols. Our
methods are based on recent results for probabilistic one-shot transformations
as well as a new asymptotic equipartition property for the projective relative
entropy.
