Some applications of hypercontractive inequalities in quantum information theory

Abstract

Hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation. In this note we discuss how hy-percontractive inequalities, in various settings, can be used to obtain (fairly) concise proofs of several results in quantum information theory: a recent lower bound of Lancien and Winter on the bias achievable by local measurements which are 4-designs; spectral concentration bounds for k-local Hamiltonians; and a recent result of Pellegrino and Seoane-Sepúlveda giving general lower bounds on the classical bias obtainable in multiplayer XOR games. 1