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.

翻译：最大相对熵及其诱导的条件最小熵已成为单次信息论的核心概念。二者均可通过正半定锥上的锥规划表示。近期研究表明，将同一锥规划改为在可分锥上定义时，可赋予其通过量子信道传输经典信息的操作解释。本研究推广了这一替换锥的框架，以确定量子信息论中哪些结果依赖于正半定锥、哪些可被泛化。我们证明：若锥的资源性指数增长但始终无法逼近正半定锥，则全量子Stein引理与渐近等分性质会失效。但对于经典量子(CQ)态，可分锥足以恢复渐近理论，从而在全量子与部分量子场景间划出显著界限。我们针对扩展条件最小熵给出了并行结果，并将k-超正信道的概念推广至超信道。此外，我们展示了该框架的操作应用：首先证明Choi算符的锥限制最小熵可刻画受限测量下的无噪声纠缠辅助经典通信容量；其次表明量子majorization结果可自然泛化至其他锥；作为新示例，我们引入类似最小熵的新型量，以双随机预处理形式捕获量子信道的量子majorization特性。最后，我们将该框架与一般锥范数及其非可加性相关联。全文突出所引入度量与一般凸资源理论的关系，特别关注刻画局域性的资源理论以及相干性/阿贝尔对称性的资源理论。