The Hilbertian Tensor Norm and Entangled Two-Prover Games

TL;DR

This paper explores the Hilbertian tensor norm's role in quantum information, establishing new inequalities and bounds that deepen understanding of entangled two-prover games and their classical and quantum value ratios.

Contribution

It introduces a generalized Hilbertian tensor norm framework for two-prover games, proving a Grothendieck inequality and providing new bounds on quantum-classical value ratios.

Findings

Proved a generalized Grothendieck inequality for tensor norms.

Provided an alternative proof of the parallel repetition theorem for entangled XOR games.

Established a new upper bound on the ratio of quantum to classical game values.

Abstract

We study tensor norms over Banach spaces and their relations to quantum information theory, in particular their connection with two-prover games. We consider a version of the Hilbertian tensor norm $\gamma_2$ and its dual $\gamma_2^*$ that allow us to consider games with arbitrary output alphabet sizes. We establish direct-product theorems and prove a generalized Grothendieck inequality for these tensor norms. Furthermore, we investigate the connection between the Hilbertian tensor norm and the set of quantum probability distributions, and show two applications to quantum information theory: firstly, we give an alternative proof of the perfect parallel repetition theorem for entangled XOR games; and secondly, we prove a new upper bound on the ratio between the entangled and the classical value of two-prover games.