Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup and determining concentration properties of its invariant state have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincaré inequalities) and the so-called transportation cost inequalities have proved to be essential for this purpose. Classical functional and transportation cost inequalities are seen to arise from a single geometric inequality, called the Ricci lower bound, via an inequality which interpolates between them. The latter is called the HWI inequality, where the letters I, W and H are, respectively, acronyms for the Fisher information (arising in the modified logarithmic Sobolev inequality), the so-called Wasserstein distance (arising in the transportation cost inequality) and the relative entropy (or Boltzmann H function) arising in both. Hence, classically, the above inequalities and the implications between them form a remarkable picture which relates elements from diverse mathematical fields, such as Riemannian geometry, information theory, optimal transport theory, Markov processes, concentration of measure and convexity theory. Here, we consider a quantum version of the Ricci lower bound introduced by Carlen and Maas and prove that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow. Our results hence establish that the unifying picture of the classical setting carries over to the quantum one.

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相对熵，最优输运和费舍尔信息：量子HWI不等式

量子马尔可夫半群表征一类重要的开放量子系统的时间演化。研究这种半群的收敛性质并确定其不变态的浓度性质已成为许多研究的重点。功能不等式的量子形式（如修正的对数Sobolev和Poincaré不等式）和所谓的运输成本不等式已被证明对于此目的至关重要。经典的功能和运输成本不平等被认为是由一个几何不平等（称为里奇下界）引起的，这种不平等是通过在不等式之间进行插值而产生的。后者称为HWI不等式，其中字母I，W和H分别是Fisher信息的缩写（在修正的对数Sobolev不等式中出现），所谓的Wasserstein距离（导致运输成本不平等加剧）和相对熵（或Boltzmann H函数）两者都产生。因此，经典地讲，上述不等式及其之间的含意形成了一张令人印象深刻的图片，该图片涉及来自多个数学领域的元素，如黎曼几何，信息论，最优输运理论，马尔可夫过程，测度集中度和凸度理论。在这里，我们考虑由卡伦（Carlen）和马斯（Maas）提出的里奇下界的量子形式，并证明它暗示着量子HWI不等式，由此可以得出量子功能和运输成本的不等式。因此，我们的结果证明，经典背景的统一图景可以延续到量子点。