Some applications of hypercontractive inequalities in quantum information theory

TL;DR

This paper explores how hypercontractive inequalities serve as powerful tools in quantum information theory, providing concise proofs for bounds on measurement bias, spectral concentration, and classical bias in multiplayer games.

Contribution

It demonstrates the application of hypercontractive inequalities to derive simplified proofs of key results in quantum information theory.

Findings

Lower bound on bias achievable by 4-design measurements

Spectral concentration bounds for k-local Hamiltonians

Lower bounds on classical bias in multiplayer XOR games

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 hypercontractive 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-Sepulveda giving general lower bounds on the classical bias obtainable in multiplayer XOR games.