Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of entropy is captured by the family of Rényi entropies, parameterized by a real number $α$. Conditional entropy extends the notion of entropy by quantifying uncertainty from the viewpoint of an observer with access to potentially correlated side information. However, despite their significance and the emergence of various useful definitions, a complete characterization of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterization of conditional entropy, defined through a set of axioms that are essential for any operationally meaningful definition: additivity for independent random variables, invariance under relabeling, and monotonicity under conditional mixing channels. We prove that the most general form of conditional entropy is captured by a family of measures that are exponential averages of Rényi entropies of the conditioned distribution and parameterized by a real parameter and a probability measure on the positive reals. Finally, we show that these quantities determine the rate of transformation under conditional mixing and provide a set of second laws of quantum thermodynamics with side information for states diagonal in the energy eigenbasis.

翻译：熵是衡量不确定性的基本度量，在信息论与统计学中具有核心重要性，并广泛应用于所有定量科学领域。在一组自然的操作公理下，熵的最一般形式由Rényi熵族所刻画，该族由一个实数$α$参数化。条件熵通过从具备潜在相关辅助信息的观察者视角量化不确定性，扩展了熵的概念。然而，尽管条件熵具有重要意义且已涌现出多种实用定义，满足自然操作公理集的条件熵度量的完整刻画问题始终悬而未决。本研究给出了条件熵的完整刻画，其定义基于一组对任何具有操作意义的定义都至关重要的公理：独立随机变量的可加性、重标定下的不变性，以及条件混合信道下的单调性。我们证明，条件熵的最一般形式由一个度量族所刻画，这些度量是条件分布的Rényi熵的指数平均，并由一个实参数和一个正实数上的概率测度进行参数化。最后，我们证明这些量决定了条件混合下的转换速率，并为能量本征基对角态提供了具有辅助信息的量子热力学第二定律集合。