The thermodynamic resourcefulness of quantum channels primarily depends on their underlying causal structure and their ability to generate quantum correlations. We quantify this interplay within the resource theory of athermality for bipartite quantum channels in the presence of a side channel acting as memory, referred to as the resource theory of conditional athermality. For channels with trivial output Hamiltonians, we characterize the optimal one-shot rates for distilling the identity gate from a given channel, as well as the cost of simulating the channel using the identity gate, under conditional Gibbs-preserving superchannels. We show that these rates have a direct trade-off relation with the conditional channel entropies, attributing operational significance to signaling in quantum processes. Furthermore, we establish an equipartition property for the conditional channel min-entropy for classes of channels that are either tele-covariant or no-signaling from the non-conditioning input to the conditioning output. As a consequence, we demonstrate asymptotic reversibility of the resource theory for these channels. The asymptotic conditional athermality capacity of a tele-covariant channel is half the superdense coding capacity of its Choi state. Our work establishes the conditional channel entropy as a primitive information-theoretic concept for quantum processes, elucidating its potential for wider applications in quantum information science.

翻译：量子通道的热力学资源性质主要取决于其潜在的因果结构及其产生量子关联的能力。我们在存在作为记忆的侧通道（称为条件非热性资源理论）的条件下，针对双体量子通道的非热性资源理论中量化了这种相互作用。对于输出哈密顿量平凡的通道，我们刻画了在条件吉布斯保持超通道下，从给定通道提纯恒等门的最优单次速率，以及使用恒等门模拟该通道的成本。我们证明了这些速率与条件通道熵存在直接的权衡关系，从而赋予量子过程中信号传递以操作意义。此外，我们针对两类通道（要么满足远程协变性，要么满足从非条件输入到条件输出的无信号条件）建立了条件通道最小熵的等分性质。作为推论，我们证明了这些通道的资源理论渐近可逆性。远程协变通道的渐近条件非热性容量等于其Choi态超密编码容量的一半。我们的工作将条件通道熵建立为量子过程的基本信息论概念，阐明了其在量子信息科学中更广泛应用的可能性。