We study randomized and quantum efficiency lower bounds in communication complexity. These arise from the study of zero-communication protocols in which players are allowed to abort. Our scenario is inspired by the physics setup of Bell experiments, where two players share a predefined entangled state but are not allowed to communicate. Each is given a measurement as input, which they perform on their share of the system. The outcomes of the measurements should follow a distribution predicted by quantum mechanics; however, in practice, the detectors may fail to produce an output in some of the runs. The efficiency of the experiment is the probability that neither of the detectors fails. When the players share a quantum state, this leads to a new bound on quantum communication complexity (eff*) that subsumes the factorization norm. When players share randomness instead of a quantum state, the efficiency bound (eff), coincides with the partition bound of Jain and Klauck. This is one of the strongest lower bounds known for randomized communication complexity, which subsumes all the known combinatorial and algebraic methods including the rectangle (corruption) bound, the factorization norm, and discrepancy. The lower bound is formulated as a convex optimization problem. In practice, the dual form is more feasible to use, and we show that it amounts to constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for eff*). For one-way communication, we show that the quantum one-way partition bound is tight for classical communication with shared entanglement up to arbitrarily small error.
@cite_19: We show that the detection efficiencies required for closing the detection loophole in Bell tests can be significantly lowered using quantum systems of dimension larger than two. We introduce a series of asymmetric Bell tests for which an efficiency arbitrarily close to 1 N can be tolerated using N-dimensional systems, and a symmetric Bell test for which the efficiency can be lowered down to 61.8 using four-dimensional systems. Experimental perspectives for our schemes look promising considering recent progress in atom-photon entanglement and in photon hyperentanglement.
V 'ertesi show that there is a distribution with boolean outputs @math , based on partially entangled states, such that (in our language) @math @cite_19 . Therefore, our results imply that @math . Since the states are nearly separable, however @math for large enough @math .