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 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.

18 pages; v3: historical and typo fixes and additional remarks

References in corpus (4)

• Unbounded violation of tripartite Bell inequalities
• Remote preparation of quantum states
• Quantum logarithmic Sobolev inequalities and rapid mixing
• Distinguishing multi-partite states by local measurements

Cited by in corpus (14)

• Hypercontractivity of quasi-free quantum semigroups
• Distinguishing multi-partite states by local measurements
• Optimal Frobenius light cone in spin chains with power-law interactions
• Extremal eigenvalues of local Hamiltonians
• Quantum reverse hypercontractivity
• Average-case Speedup for Product Formulas
• Towards sharp Bohnenblust--Hille constants
• Contractivity properties of a quantum diffusion semigroup
• Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles
• The optimal multilinear Bohnenblust-Hille constants: a computational solution for the real case
• Fourier 1-norm and quantum speed-up
• Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case
• The optimal constants for the real Hardy--Littlewood inequality for bilinear forms on $c_0\times {\ell }_p$
• The sharp constants in the real anisotropic Littlewood's $4/3$ inequality and applications