A colloquial interpretation of entropy is that it is the knowledge gained upon learning the outcome of a random experiment. Conditional entropy is then interpreted as the knowledge gained upon learning the outcome of one random experiment after learning the outcome of another, possibly statistically dependent, random experiment. In the classical world, entropy and conditional entropy take only non-negative values, consistent with the intuition that one has regarding the aforementioned interpretations. However, for certain entangled states, one obtains negative values when evaluating commonly accepted and information-theoretically justified formulas for the quantum conditional entropy, leading to the confounding conclusion that one can know less than nothing in the quantum world. Here, we introduce a physically motivated framework for defining quantum conditional entropy, based on two simple postulates inspired by the second law of thermodynamics (non-decrease of entropy) and extensivity of entropy, and we argue that all plausible definitions of quantum conditional entropy should respect these two postulates. We then prove that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world. All of our arguments are based on constructions of physical processes that respect the first postulate, the one inspired by the second law of thermodynamics.

中文翻译：

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知之少的必然性


熵的口语解释是，它是在了解随机实验结果后获得的知识。然后，条件熵被解释为在学习一个随机实验的结果后，在学习另一个随机实验的结果后获得的知识，这可能依赖于统计。在古典世界中，熵和条件熵只取非负值，这与人们对上述解释的直觉一致。然而，对于某些纠缠态，在评估普遍接受的和信息论上合理的量子条件熵公式时，人们会得到负值，从而导致令人困惑的结论，即在量子世界中，人们所能知道的比什么都少。在这里，我们引入了一个物理驱动的框架来定义量子条件熵，该框架基于受热力学第二定律（熵的不减少）和熵的广延性启发的两个简单假设，我们认为量子条件熵的所有合理定义都应该尊重这两个假设。然后，我们证明所有合理的量子条件熵对于某些纠缠状态都为负值，因此在量子世界中，人们不可避免地会知道比什么都少。我们所有的论点都是基于尊重第一假设的物理过程的构造，该假设受到热力学第二定律的启发。