Generalized XOR games with $d$ outcomes and the task of non-local computation

TL;DR

This paper generalizes XOR games to multiple outcomes, providing algebraic bounds on quantum values, and explores the limitations of quantum advantage in non-local computation for prime outcomes, revealing fundamental constraints.

Contribution

It introduces an algebraic bound for quantum values of XOR-d games and extends the principle of no quantum advantage to prime-d outcome functions in non-local computation.

Findings

Quantum strategies do not outperform classical ones in uniform-input XOR-d games.

No total function XOR-d game with uniform inputs can be a pseudo-telepathy game.

Binary non-local computation inequalities are not facet defining for any number of inputs.

Abstract

A natural generalization of the binary XOR games to the class of XOR-d games with $d > 2$ outcomes is studied. We propose an algebraic bound to the quantum value of these games and use it to derive several interesting properties of these games. As an example, we re-derive in a simple manner a recently discovered bound on the quantum value of the CHSH-d game for prime $d$. It is shown that no total function XOR-d game with uniform inputs can be a pseudo-telepathy game, there exists a quantum strategy to win the game only when there is a classical strategy also. We then study the principle of lack of quantum advantage in the distributed non-local computation of binary functions which is a well-known information-theoretic principle designed to pick out quantum correlations from amongst general no-signaling ones. We prove a large-alphabet generalization of this principle, showing that quantum theory provides no advantage in the task of non-local distributed computation of a restricted class of functions with $d$ outcomes for prime $d$, while general no-signaling boxes do. Finally, we consider the question whether there exist two-party tight Bell inequalities with no quantum advantage, and show that the binary non-local computation game inequalities for the restricted class of functions are not facet defining for any number of inputs.