  The max-relative entropy and the conditional min-entropy it induces have
become central to one-shot information theory. Both may be expressed in terms
of a conic program over the positive semidefinite cone. Recently, it was shown
that the same conic program altered to be over the separable cone admits an
operational interpretation in terms of communicating classical information over
a quantum channel. In this work, we generalize this framework of replacing the
cone to determine which results in quantum information theory rely upon the
positive semidefinite cone and which can be generalized. We show the fully
quantum Stein's lemma and asymptotic equipartition property break down if the
cone exponentially increases in resourcefulness but never approximates the
positive semidefinite cone. However, we show for CQ states, the separable cone
is sufficient to recover the asymptotic theory, thereby drawing a strong
distinction between the fully and partial quantum settings. We present parallel
results for the extended conditional min-entropy. In doing so, we extend the
notion of k-superpositive channels to superchannels. We also present
operational uses of this framework. We first show the cone restricted
min-entropy of a Choi operator captures a measure of entanglement-assisted
noiseless classical communication using restricted measurements. We show that
quantum majorization results naturally generalize to other cones. As a novel
example, we introduce a new min-entropy-like quantity that captures the quantum
majorization of quantum channels in terms of bistochastic pre-processing.
Lastly, we relate this framework to general conic norms and their
non-additivity. Throughout this work we emphasize the introduced measures'
relationship to general convex resource theories. In particular, we look at
both resource theories that capture locality and resource theories of
coherence/Abelian symmetries.
