On the relation between completely bounded and $(1,cb)$-summing maps with applications to quantum XOR games

TL;DR

This paper establishes a bound relating completely bounded and $(1,cb)$-summing norms for linear maps into dual C*-algebras, with implications for quantum XOR games and the limitations of entangled strategies.

Contribution

It proves a universal bound connecting two operator space norms for maps into dual C*-algebras, impacting quantum information theory.

Findings

Bound on the completely bounded norm in terms of the $(1,cb)$-summing norm

Implication that entangled strategies in quantum XOR games are limited

Provides tools for analyzing quantum strategies using operator space theory

Abstract

In this work we show that, given a linear map from a general operator space into the dual of a C$^*$-algebra, its completely bounded norm is upper bounded by a universal constant times its $(1,cb)$-summing norm. This problem is motivated by the study of quantum XOR games in the field of quantum information theory. In particular, our results imply that for such games entangled strategies cannot be arbitrarily better than those strategies using one-way classical communication.