We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL Theorem, Friedgut's Junta Theorem and Talagrand's variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities and the learnability of quantum observables.

翻译：我们将分析布林功能影响得出的三个相关结果推广到量子设置,即KKL Theorem、Friedgut的Junta Theorem和Talagrand的几何影响差异不平等。我们的结果来自最近研究过的超分率和梯度估计数的共同使用。这些通用工具还使我们能够在量子超立方外的Von Neumann代数法中得出这些结果的概括性,包括与量子信息理论有关的无限层面的例子,例如连续变量量子系统。最后,我们评论了我们的结果对异差类型不平等的非平衡扩展和可观测量子的可学性的影响。