We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a,b distributed according to some pre-specified joint distribution p(a,b|x,y). We introduce a new technique based on affine combinations of lower-complexity distributions. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. These lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication.
Communication complexity of Boolean functions has a long and rich past, stemming from the paper of Yao in 1979 [Yao79], whose motivation was to study the area of VLSI circuits. In the years that followed, tremendous progress has been made in developing a rich array of lower bound techniques for various models of communication complexity (see e.g. [KN97]).
From the physics side, the question of studying how much communication is needed to simulate distributions arising from physical phenomena, such as measuring bipartite quantum states, was posed in 1992 by Maudlin, a philosopher of science, who wanted to quantify the non-locality inherent to these systems [Mau92]. Maudlin, and the authors who followed [BCT99, Ste00, TB03, CGMP05, DLR07] (some independently of his work, and of each other) progressively improved upper bounds on simulating correlations of the 2 qubit singlet state. In a recent breakthrough, Regev and Toner [RT10] proved that two bits of communication suffice to simulate the correlations arising from two-outcome measurements of arbitrary-dimension bipartite quantum states. In the more general case of non-binary Finally, in Section 6, we give upper bounds on simultaneous messages complexity in terms of our lower bound expression (Theorem 8). We use fingerprinting methods [BCWdW01,Yao03,SZ08,GKd06] to give very simple proofs that classical communication with shared randomness, or quantum communication with shared entanglement, can be simulated in the simultaneous messages model, with exponential blowup in communication, and in particular that any quantum distribution can be approximated with constant communication.
The use of affine combinations for non-signaling distributions has roots in the quantum logic community, where quantum non-locality has been studied within the setting of more general probability theories [FR81,RF81,KRF87,Wil92]. Until recently, this line of work was largely unknown in the quantum information theory community [Bar07,BBLW07].
The structure of the non-signaling polytope has been the object of much study. A complete characterization of the vertices has been obtained in some, but not all cases: for two players, the case of binary inputs [BLM + 05], and the case of binary outputs [BP05,JM05] are known, and for n players, the case of Boolean inputs and outputs is known [BP05].
The work on simulating quantum distributions has focused mainly on providing upper bounds, and most results apply to simulating the correlations only. In particular, Toner and Bacon show that projective measurements on a maximally entangled qubit pair may be simulated using one bit of communication [TB03], and Regev and Toner extend this result by showing that the correlations arising from binary measurements on any entangled state may be simulated using two bits of communication only [RT10]. A few results address the simulation of quantum distributions with non-uniform marginals. Bacon and Toner give an upper bound of 2 bits for non-maximally entangled qubit pairs [TB03]. Shi and Zhu [SZ08] show a constant upper bound for approximating any quantum distribution (including the marginals) to within a constant.
Pironio gives a general lower bound technique based on Bell-like inequalities [Pir03]. There are a few ad hoc lower bounds on simulating quantum distributions, including a linear lower bound for a distribution based on Deutsch-Jozsa’s problem [BCT99], and a recent lower bound of Gavinsky [Gav09].
The γ 2 method was first introduced as a measure of the complexity of matrices [LMSS07]. It was shown to be a lower bound on communication complexity [LS09], and to generalize many previously known methods. Lee et al. use it to establish direct product theorems and relate the dual norm of γ 2 to the value of XOR games [LS Š08]. Lee and Shraibman [LS08] use a multidimensional generalization of a related quantity µ (where the norm-1 ball consists of cylinder intersections) to prove a lower bound in the multiparty number-on-the-forehead-model, for the disjointness function.
Since the first publication of this work, several extensions and improvements have been made to the upper bounds on Bell inequality violations of Section 5, and related lower bounds on the possible violations have been proved [JPPG + 10b, JPPG + 10a, JP10, BRSdW10].
In this paper, we extend the framework of communication complexity to non-signaling distributions. This framework encompasses the standard models of communication complexity of Boolean functions but also total and partial non-Boolean functions and relations, as well as distributions arising from the measurements of bipartite quantum states. Most results we present also extend to the multipartite setting.
Throughout this article, we consider bipartite conditional distributions p(a, b|x, y) where x ∈ X , y ∈ Y are the inputs of the players, and they are required to each produce an outcome a ∈ A, b ∈ B, distributed according to p(a, b|x, y). We will focus on so-called non
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