The quantum data processing inequality asserts that two quantum states become harder to distinguish when a noisy channel is applied. On the other hand, a reverse quantum data processing inequality characterizes whether distinguishability is preserved after the application of a noisy channel. In this work, we explore these concepts through contraction and expansion coefficients of the relative entropy of quantum channels. Our first result is that quantum channels with an input dimension greater than or equal to the output dimension do not have a non-zero expansion coefficient, which means that they cannot admit a reverse data-processing inequality. We propose a comparative approach by introducing a relative expansion coefficient, to assess how one channel expands relative entropy compared to another. We show that this relative expansion coefficient is positive for three important classes of quantum channels: depolarizing channels, generalized dephasing channels, and amplitude damping channels. As an application, we give the first rigorous construction of level-1 less noisy quantum channels that are non-degradable.

翻译：量子数据处理不等式断言，当施加一个噪声信道时，两个量子态将变得更难区分。另一方面，反向量子数据处理不等式则刻画了在施加噪声信道后，可区分性是否得以保持。在本工作中，我们通过量子信道相对熵的收缩系数与扩张系数来探讨这些概念。我们的第一个结果是：输入维度大于或等于输出维度的量子信道不具有非零的扩张系数，这意味着它们不可能满足反向数据处理不等式。我们提出了一种比较方法，通过引入相对扩张系数来评估一个信道相对于另一个信道扩张相对熵的能力。我们证明，对于三类重要的量子信道：退极化信道、广义退相信道和振幅阻尼信道，该相对扩张系数为正。作为应用，我们首次严格构造了非可降解的1级更安静量子信道。