Quantum extensions of Talagrand, KKL and Freidgut’s theorems

In 2008, Montanaro and Osborne [MO] proposed a quantum extension of Boolean functions, namely self-adjoint unitary matrices on $(\mathbb{C}^2)^{\otimes n}$, and extended the celebrated Talagrand L1-L2 variance inequality to this setting. In the classical case, the latter was shown to imply the so-called KKL theorem about the existence of an influential variable for a Boolean function as a simple corollary. However, since the L1 influences and L2 influences do not agree for general quantum Boolean functions, the extension of [MO] does not provide the quantum KKL theorem, in sharp contrast with the classical setting. Therefore a quantum version of the KKL theorem is still missing and was kept as a conjecture in [MO]. In this talk, I will argue that every balanced quantum Boolean function has a geometrically influential variable. I will also derive a quantum analogue of the related Friedgut junta theorem about the approximation of Boolean functions of small total influence by functions of few variables. These results are based on the joint use of recently studied hypercontractivity and gradient estimates. Such 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. If time permits, I will comment on the implications of our results as regards to noncommutative extensions of isoperimetric inequalities and the learnability of quantum observables. This is based on joint work with Melchior Wirth and Haonan Zhang (IST Austria).