Heisenberg's uncertainty principle, which imposes intrinsic restrictions on our ability to predict the outcomes of incompatible quantum measurements to arbitrary precision, demonstrates one of the key differences between classical and quantum mechanics. The physical systems considered in the uncertainty principle are static in nature and described mathematically with a quantum state in a Hilbert space. However, many physical systems are dynamic in nature and described with the formalism of a quantum channel. In this paper, we show that the uncertainty principle can be reformulated to include process measurements that are performed on quantum channels. Since both the preparation of quantum states and the implementation of quantum measurements are themselves special cases of quantum channels, our formalism encapsulates the uncertainty principle in its utmost generality. More specifically, we obtain expressions that generalize the Maassen–Uffink uncertainty relation and the universal uncertainty relations from quantum states to quantum channels.

中文翻译：

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量子过程的不确定性原理

海森堡的不确定性原则对我们预测不相容的量子测量结果达到任意精度的能力施加了固有的限制，这证明了经典力学和量子力学之间的主要区别之一。在不确定性原理中考虑的物理系统本质上是静态的，并用希尔伯特空间中的量子态进行数学描述。但是，许多物理系统本质上是动态的，并以量子通道的形式描述。在本文中，我们表明不确定性原理可以重新制定，以包括在量子通道上执行的过程测量。由于准备量子态和执行量子测量本身都是量子通道的特殊情况，我们的形式主义将不确定性原则以其最大的概括性封装起来。更具体地说，我们得到的表达式可以推广从量子态到量子通道的Maassen-Uffink不确定性关系和普遍不确定性关系。