Quantum communication complexity advantage implies violation of a Bell inequality

Abstract

We obtain a general connection between a large quantum advantage in communication complexity and Bell nonlocality. We show that given any protocol offering a sufficiently large quantum advantage in communication complexity, there exists a way of obtaining measurement statistics that violate some Bell inequality. Our main tool is port-based teleportation. If the gap between quantum and classical communication complexity can grow arbitrarily large, the ratio of the quantum value to the classical value of the Bell quantity becomes unbounded with the increase in the number of inputs and outputs.

Classical versus quantum communication in XOR games

“…We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient ω * o.w.−c (G)/ωo.w.−2c(G) is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al (2016).We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information c. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical vs quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.We will refer to one such B as a Bell functional.…”

“…The randomized communication complexity of f is defined to be the minimum number of bits (or qubits in the quantum case) interchanged between Alice and Bob required for a randomized algorithm in order to compute correctly f (x, y) with probability larger than ǫ for every possible input (x, y). We will call these numbers CC(f, ǫ) and QC(f, ǫ) respectively.In this paper we study the relation between Bell inequality violations and communication complexity problems [3], continuing the spirit of the recent paper [4], where some new implications between both contexts where uncovered. In this line, certain specific Bell inequality violations are known to lead to separation in communication complexity for certain functions [2], although we do not know of any general implication in this direction.…”

“…In this line, certain specific Bell inequality violations are known to lead to separation in communication complexity for certain functions [2], although we do not know of any general implication in this direction. For the other direction, a recent result [4] shows that it is possible in general to obtain Bell inequality violations starting from large enough separations in communication complexity. Some other interesting results relating communication complexity and Bell inequality violations have been recently obtained in [10].…”

Randomness and Non-Locality

“…In parallel, other quantum protocols offering a communication complexity advantage were being discovered [18,42,43] for which, although suspected, it was not clear whether non-locality or other future of quantum mechanics was the responsible for the improved communication efficiency. In a recent breakthrough, it has been shown by Buhrman et al [44] that, indeed, non-locality is the key future of quantum mechanics responsible for the advantages in communication complexity. More formally, they have shown that for every communication complexity problem for which there is a quantum protocol achieving a (greater than quadratic) advantage over all classical strategies with shared randomness, there is a Bell inequality and a quantum distribution that violates it.…”

Large bipartite Bell violations with dichotomic measurements

“…It is natural that large violations of Bell inequalities provide more benefit, both in theory and applications. In fact, the study of the asymptotic behavior of Bell violations is essential to understand this phenomenon, and the results along these lines have been very useful to learn about certain properties of quantum nonlocality and its relation with other resources [13][14][15][16].…”