Exponential Separation of Quantum Communication and Classical Information

TL;DR

This paper demonstrates an exponential separation between quantum communication complexity and classical information complexity for a specific Boolean function, revealing their fundamental incomparability and implications for quantum communication theory.

Contribution

It provides the first exponential separation showing quantum communication complexity can be exponentially larger than classical information complexity for a specific function.

Findings

Quantum communication complexity is polynomially equivalent to classical communication complexity for the Symmetric k-ary Pointer Jumping function.

Classical information complexity is an upper bound on quantum information complexity, but they are incomparable.

A simple proof of the optimal trade-off in the Greater-Than function communication complexity, even with entanglement.

Abstract

We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha. As another application of the techniques that we develop, we give a simple proof for an optimal trade-off between Alice's and Bob's communication while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when allowing pre-shared entanglement. We also present a classical protocol achieving this bound.