We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation–dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory.

我们通过研究显示各种有趣现象和奇点的量子仪器实例，说明了在先前工作中发展的重复量子测量过程中熵产生的数学理论。我们强调了热力学形式主义的作用，并给出了许多量子仪器的例子，这些量子仪器对结果的无限序列空间（移位空间）的概率度量不具有（弱）吉布斯性质。我们还讨论了物理上相关的示例，这些示例中的熵产生率满足大偏差原理，但没有服从中心极限定理和波动耗散定理。在整个分析过程中，我们探索了与其他先验无关主题的联系，例如马尔可夫链的功能，隐马尔可夫模型，矩阵乘积和数论。