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.

翻译：熵的一种通俗解释是：它是在获知随机实验结果时所获得的知识量。条件熵则被解释为，在已知另一个可能具有统计依赖性的随机实验结果后，再获知该随机实验结果时所获得的知识量。在经典世界中，熵和条件熵仅取非负值，这与上述解释所引发的直觉相符。然而，对于某些纠缠态，当评估公认且在信息论上合理的量子条件熵公式时，会得到负值，从而得出一个令人困惑的结论：在量子世界中，认知可能少于零。本文引入一个基于物理动机的框架来定义量子条件熵，该框架受热力学第二定律（熵的非递减性）和熵的广延性启发，提出了两个简单公设，并论证所有合理的量子条件熵定义都应遵循这两个公设。随后，我们证明对于某些纠缠态，所有合理的量子条件熵均会取负值，因此在量子世界中认知少于零是不可避免的。我们所有的论证均基于遵循第一个公设（即受热力学第二定律启发的公设）的物理过程构造。