In this work we investigate Stinespring dilations of quantum-dynamical semigroups, which are known to exist by means of a constructive proof given by Davies in the early 70s. We show that if the semigroup describes an open system, that is, if it does not consist of only unitary channels, then the evolution of the dilated closed system has to be generated by an unbounded Hamiltonian; subsequently the environment has to correspond to an infinite-dimensional Hilbert space, regardless of the original system. Moreover, we prove that the second derivative of Stinespring dilations with a bounded total Hamiltonian yields the dissipative part of some quantum-dynamical semigroup — and vice versa. In particular this characterizes the generators of quantum-dynamical semigroups via Stinespring dilations.

在这项工作中，我们研究了量子动力学半群的 Stinespring 膨胀，已知这些膨胀是通过戴维斯在 70 年代初期给出的建设性证明而存在的。我们表明，如果半群描述一个开放系统，也就是说，如果它不仅仅由酉通道组成，那么扩张封闭系统的演化必须由无界哈密顿量产生；随后，无论原始系统如何，环境都必须对应于一个无限维的希尔伯特空间。此外，我们证明了具有有界总哈密顿量的 Stinespring 扩张的二阶导数产生了某些量子动力学半群的耗散部分，反之亦然。特别是，这通过 Stinespring 膨胀表征了量子动力学半群的生成元。