arXiv:1308.5404v1 [quant-ph] 25 Aug 2013 Universal scheme for violation of local realism from quantum advantage in one-way communication complexity L. Czekaj, 1 A. Grudka, 2 M. Horodecki, 1 P. Horodecki, 3 and M. Markiewicz 1 1 Faculty of Mathematics, Physics and Informatics, Gda´ nsk University, 80-952 Gda´ nsk,Poland 2 Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´ n, Poland 3 Faculty of Applied Physics and Mathematics, Gda´ nsk University of Technology, 80-952 Gda´ nsk, Poland We consider relations between communication complexity problems and detecting correlations (violating local realism) with no local hidden variable model. We show first universal equivalence between characteristics of protocols used in that type of problems and non-signaling correlations. We construct non linear bipartite Bell type inequalities and strong nonlocality test with binary observables by providing general method of Bell inequalities construction and showing that existence of gap between quantum and classical complexity leads to violation of these inequalities. We obtain, first to our knowledge, explicit Bell inequality with binary observables and exponential violation. PACS numbers: Key element which distinguishes classical and quan- tum world are quantum correlations. The strength of these correlations was first realized in EPR para- dox [1] and then quantitatively expressed in Bell the- orem [2]. Although non-signaling, they cannot be re- duced to local hidden variable model. This property leads to another approach to understanding quantum correlations, where they are taken as a resource which abridges ”hardness” of certain information processing tasks [3–5]. Reduction of communication cost of solv- ing certain distributed computational problems by use of quantum correlations is the result on this ground which emphasizes non-local character of quantum cor- relations [6, 7]. For a long time violation of local realism (i.e. ex- istence of correlations that cannot be described by local hidden variable model) was of interest of philo- sophically oriented physicists, and was considered as a kind of exotic peculiarity, that does not affect our life. Quantum information era has completely reversed this picture: local realism and its quantum mechanical vi- olation has become a practical resource. One area, where quantum mechanical violation has practical im- plications is communication complexity. An everyday task is to compute some function of distributed argu- ments. For example, ”doodle” utility allows to sched- ule appointment for distant parties. Now the question is: how much information needs to be exchanged to find the time slot which will be suitable for all parties? The amount of bits needed to exchange in order to solve some common problem is called communica- tion complexity. Restricting to two parties (Alice and Bob), we consider a function f (x,y), such that Alice has x and Bob has y. We assume some apriori distri- bution over x,y, and ask about communication com- plexity, i.e. the number of bits needed to exchange in order to compute the function by e.g. Bob. It turns out that if Alice and Bob share an entangled state they may need to exchange much less bits, than when they share classical correlations (aka shared random- ness) [3, 8]. Intuitively this means, that the statistics of outcomes of measurement performed on such state must violate local realism. Indeed, if it were possi- ble to describe the statistics in a local realistic way, this would mean that instead of having such entan- gled state, we might have used a local realistic model, which is nothing more than classical shared random- ness. Therefore, the number of bits could not have been smaller than needed in classical case. Putting it in a different way: if the results of measurement exist prior to the measurement, equally well, the experi- ment could be simulated by writing those preexisting values on a piece of paper, and then simply reading them out. This intuition is confirmed by many examples: first protocols of quantum advantage were based on known earlier examples of violation of local realism, mani- fested by violation of Bell inequalities [3]. Moreover it was shown that all Bell inequalities of certain type lead to a quantum advantage (for a perhaps peculiar function [9]). Instead of sharing entanglement, Alice and Bob might be allowed to transmit qbits. For this scheme the quantum advantage (i.e. that one needs to send considerable less amount of qbits than bits) is also manifested for some functions. Such a scheme can always be converted to the scheme, where the par- ties send bits, and share entangled state. Most pro- found is here the famous Raz protocol [7, 10], where the quantum advantage over classical communication complexity is exponential. Another prominent exam- ple is so called ”hidden matching”, which served to obtain superstrong violation of Bell inequality [11]. However, despite the clear intuition, there is no uni- versal protocol of the following sort: given quantum advantage in communication complexity, provide a vi- olation of some Bell inequality. The mentioned exam- ple of Bell inequality obtained from ”hidden match- ing” is based on some special symmetries. Also a general theorem, which says that quantum advantage implies violation of local realism, requires some very particular symmetries of the quantum protocol [6].