Robust Bell Inequalities from Communication Complexity

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 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 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. 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 bounds. 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

Entanglement and communication-reducing properties of noisy N-qubit states

We consider properties of states of many qubits, which arise after sending certain entangled states via various noisy channels (white noise, coloured noise, local depolarization, dephasing and amplitude damping). Entanglement of these states is studied and their ability to violate certain classes of Bell inequalities. States which violate them allow for higher than classical efficiency of solving related distributed computational tasks with constrained communication. This is a direct property of such states -- not requiring their further modification via stochastic local operations and classical communication such as entanglement purification or distillation procedures. We identify novel families of multi-particle states which are entangled but nevertheless allow local realistic description of specific Bell experiments. For some of them, the "gap" between the critical values for entanglement and violation of Bell inequality remains finite even in the limit of infinitely many qubits.Comment: new version, more results adde

Multipartite quantum nonlocality under local decoherence

We study the nonlocal properties of two-qubit maximally-entangled and N-qubit Greenberger-Horne-Zeilinger states under local decoherence. We show that the (non)resilience of entanglement under local depolarization or dephasing is not necessarily equivalent to the (non)resilience of Bell-inequality violations. Apart from entanglement and Bell-inequality violations, we consider also nonlocality as quantified by the nonlocal content of correlations, and provide several examples of anomalous behaviors, both in the bipartite and multipartite cases. In addition, we study the practical implications of these anomalies on the usefulness of noisy Greenberger-Horne-Zeilinger states as resources for nonlocality-based physical protocols given by communication complexity problems. There, we provide examples of quantum gains improving with the number of particles that coexist with exponentially-decaying entanglement and non-local contents.Comment: 6 pages, 4 figure

Classical and quantum partition bound and detector inefficiency

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 the experiment succeeds (neither of the detectors fails). When the players share a quantum state, this gives rise 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*). We give an example of a quantum distribution where the violation can be exponentially bigger than the previously studied class of normalized Bell inequalities. 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.Comment: 21 pages, extended versio

Bell nonlocality

Bell's 1964 theorem, which states that the predictions of quantum theory cannot be accounted for by any local theory, represents one of the most profound developments in the foundations of physics. In the last two decades, Bell's theorem has been a central theme of research from a variety of perspectives, mainly motivated by quantum information science, where the nonlocality of quantum theory underpins many of the advantages afforded by a quantum processing of information. The focus of this review is to a large extent oriented by these later developments. We review the main concepts and tools which have been developed to describe and study the nonlocality of quantum theory, and which have raised this topic to the status of a full sub-field of quantum information science.Comment: 65 pages, 7 figures. Final versio

A Bell-type test of energy-time entangled qutrits

We have performed a Bell-type test for energy-time entangled qutrits. A method of inferring the Bell violation in terms of an associated interference visibility is derived. Using this scheme we obtained a Bell value of $2.784 \pm 0.023$, representing a violation of $34 \sigma$ above the limit for local variables. The scheme has been developed for use at telecom wavelengths and using proven long distance quantum communication architecture to optimize the utility of this high dimensional entanglement resource.Comment: replaced lost acknowledement

Bell scenarios with communication

Classical and quantum physics provide fundamentally different predictions about experiments with separate observers that do not communicate, a phenomenon known as quantum nonlocality. This insight is a key element of our present understanding of quantum physics, and also enables a number of information processing protocols with security beyond what is classically attainable. Relaxing the pivotal assumption of no communication leads to new insights into the nature quantum correlations, and may enable new applications where security can be established under less strict assumptions. Here, we study such relaxations where different forms of communication are allowed. We consider communication of inputs, outputs, and of a message between the parties. Using several measures, we study how much communication is required for classical models to reproduce quantum or general no-signalling correlations, as well as how quantum models can be augmented with classical communication to reproduce no-signalling correlations.Comment: 12 pages, 3 figures. Includes a more detailed explanation of results appearing in the appendix of arXiv:1411.4648 [quant-ph

Optimal measurements for nonlocal correlations

A problem in quantum information theory is to find the experimental setup that maximizes the nonlocality of correlations with respect to some suitable measure such as the violation of Bell inequalities. The latter has however some drawbacks. First and foremost it is unfeasible to determine the whole set of Bell inequalities already for a few measurements and thus unfeasible to find the experimental setup maximizing their violation. Second, the Bell violation suffers from an ambiguity stemming from the choice of the normalization of the Bell coefficients. An alternative measure of nonlocality with a direct information-theoretic interpretation is the minimal amount of classical communication required for simulating nonlocal correlations. In the case of many instances simulated in parallel, the minimal communication cost per instance is called nonlocal capacity, and its computation can be reduced to a convex-optimization problem. This quantity can be computed for a higher number of measurements and turns out to be useful for finding the optimal experimental setup. Focusing on the bipartite case, in this paper, we present a simple method for maximizing the nonlocal capacity over a given configuration space and, in particular, over a set of possible measurements, yielding the corresponding optimal setup. Furthermore, we show that there is a functional relationship between Bell violation and nonlocal capacity. The method is illustrated with numerical tests and compared with the maximization of the violation of CGLMP-type Bell inequalities on the basis of entangled two-qubit as well as two-qutrit states. Remarkably, the anomaly of nonlocality displayed by qutrits turns out to be even stronger if the nonlocal capacity is employed as a measure of nonlocality.Comment: Some typos and errors have been corrected, especially in the section concerning the relation between Bell violation and communication complexit

Experimental device-independent certified randomness generation with an instrumental causal structure

The intrinsic random nature of quantum physics offers novel tools for the generation of random numbers, a central challenge for a plethora of fields. Bell non-local correlations obtained by measurements on entangled states allow for the generation of bit strings whose randomness is guaranteed in a device-independent manner, i.e. without assumptions on the measurement and state-generation devices. Here, we generate this strong form of certified randomness on a new platform: the so-called instrumental scenario, which is central to the field of causal inference. First, we theoretically show that certified random bits, private against general quantum adversaries, can be extracted exploiting device-independent quantum instrumental-inequality violations. To that end, we adapt techniques previously developed for the Bell scenario. Then, we experimentally implement the corresponding randomness-generation protocol using entangled photons and active feed-forward of information. Moreover, we show that, for low levels of noise, our protocol offers an advantage over the simplest Bell-nonlocality protocol based on the Clauser-Horn-Shimony-Holt inequality.Comment: Modified Supplementary Information: removed description of extractor algorithm introduced by arXiv:1212.0520. Implemented security of the protocol against general adversarial attack