论文标题
Quantum Talagrand，KKL和Friedgut的定理以及量子布尔功能的可学习性
Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions
论文作者
论文摘要
我们将布尔函数影响分析的三个相关结果扩展到量子设置，即KKL定理，Friedgut的Junta定理和Talagrand的几何影响不平等。我们的结果是通过联合使用最近研究的超收缩率和梯度估计来得出的。这些通用工具还可以使我们能够在量子超立方体的情况下以一般的von Neumann代数设置来得出这些结果，其中包括与量子信息理论相关的无限维度的示例，例如连续变量量子系统。最后，我们评论结果的含义，就等型类型不平等的非交通性扩展，量子电路复杂性下限和量子可观察到的可学习性。
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.
