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, quantum circuit complexity lower bounds and the learnability of quantum observables.

翻译：我们将布尔函数影响分析的三个相关结果——即KKL定理、Friedgut的Junta定理以及Talagrand关于几何影响的方差不等式——推广至量子领域。我们的成果是通过联合运用近期发展的超压缩性和梯度估计方法而推导得出的。这些通用工具还使我们能够在量子超立方体之外的一般冯·诺依曼代数框架下推导出这些结果的推广形式，包括与量子信息论相关的无穷维例子（如连续变量量子系统）。最后，我们讨论了这些结果在等周型不等式的非交换推广、量子电路复杂度下界以及量子可观测量的可学习性方面的启示意义。