Abstract
AbstractWe 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.
Funder
Austrian Science Fund
Agence Nationale de la Recherche
Alexander von Humboldt-Stiftung
University of South Carolina
Publisher
Springer Science and Business Media LLC
Reference94 articles.
1. Atıcı, A., Servedio, R.A.: Quantum algorithms for learning and testing juntas. Quantum Inf. Process. 6(5), 323–348 (2007)
2. Alon, N., Dinur, I., Friedgut, E., Sudakov, B.: Graph products, Fourier analysis and spectral techniques. Geom. Funct. Anal. 14(5), 913–940 (2004)
3. University Lecture Series;A Auffinger,2017
4. Austin, T.: On the failure of concentration for the $$\ell ^\infty $$-ball. Israel J. Math. 211(1), 221–238 (2016)
5. Bardet, I.: Estimating the decoherence time using non-commutative functional inequalities (2017). arXiv preprint arXiv:1710.01039
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