标题： 从量子概率中涌现的几何学 显示英文标题

标题： Emergent Geometry from Quantum Probability

摘要： 将条件概率的思想推广到量子领域因其量子可观测量的非交换性而变得尤为困难。然而，在冯·诺依曼代数上的条件期望在量子信息论的发展中发挥了重要作用，尤其是在量子误差校正的研究中。在量子引力中，有人建议可以利用条件期望以代数方式实现全息图映射，其中量子误差校正在通过广义熵公式出现时空的过程中起到基础作用。然而，精确误差校正的要求对于现实的量子引力理论来说几乎肯定是过于严格了。 在这篇短文中，我们提出了一种放松版本的量子条件期望，它实现了近似误差校正。我们引入了康尼斯（Connes）空间理论的一个推广，适应于完全正映射，并推导出一个链式法则，允许相对模算符在非交换情况下分解为边缘和条件部分，构成量子贝叶斯定律。这使得可以精确量化任意量子信道的数据处理不等式中的信息差距。当应用于代数包含时，这也提供了一种将状态的熵分解为一系列项的方法，而在引力背景下，这些项可能被解释为广义熵。 我们展示了在精确误差校正的条件下，出现的面积算子是完全非交换的，而不是中心的。我们还对如何利用这一结果来构建完全代数化的量子极值曲面处方以及探测黑洞的量子性质提出了几点看法。 显示英文摘要

摘要： Carrying the insights of conditional probability to the quantum realm is notoriously difficult due to the non-commutative nature of quantum observables. Nevertheless, conditional expectations on von Neumann algebras have played a significant role in the development of quantum information theory, and especially the study of quantum error correction. In quantum gravity, it has been suggested that conditional expectations may be used to implement the holographic map algebraically, with quantum error correction underlying the emergence of spacetime through the generalized entropy formula. However, the requirements for exact error correction are almost certainly too strong for realistic theories of quantum gravity. In this note, we present a relaxed notion of quantum conditional expectation which implements approximate error correction. We introduce a generalization of Connes' spatial theory adapted to completely positive maps, and derive a chain rule allowing for the non-commutative factorization of relative modular operators into a marginal and conditional part, constituting a quantum Bayes' law. This allows for an exact quantification of the information gap occurring in the data processing inequality for arbitrary quantum channels. When applied to algebraic inclusions, this also provides an approach to factorizing the entropy of states into a sum of terms which, in the gravitational context, may be interpreted as a generalized entropy. We illustrate that the emergent area operator is fully non-commutative rather than central, except under the conditions of exact error correction. We provide some comments on how this result may be used to construct a fully algebraic quantum extremal surface prescription and to probe the quantum nature of black holes.