Abstract
The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.
Highlights
The question of achieving large Bell violations has been studied since Bell’s seminal paper in 1964 [6]
A subsequent paper [31] studied equivalent formulations of the partition bound, one of the strongest lower bounds in communication complexity [20]. This bound has several formulations: the primal formulation can be viewed as resistance to detector inefficiency, and the dual formulation is given in terms of inefficiency-resistant Bell inequality violations
We show how to obtain large inefficiency-resistant Bell violations for quantum distributions from gaps between quantum communication complexity and the efficiency bound for classical communication complexity
Summary
The question of achieving large Bell violations has been studied since Bell’s seminal paper in 1964 [6]. Buhrman et al [12] gave a construction that could be applied to several problems which had efficient quantum protocols (in terms of communication) and for which one could show a trade-off between communication and error in the classical setting. This still required an ad hoc analysis of communication problems. Buhrman et al [13] proposed the first general construction of quantum states along with Bell inequalities from any communication problem. We give upper bounds on their construction in terms of the parameters d, N, K
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