Nonlocality in the No-Signaling Framework

The behaviors that obey the no-signaling condition are a natural set to study nonlocality: this chapter is devoted to it. First, the most famous example of a no-signaling non-quantum behavior is introduced, the so-called Popescu-Rohrlich (PR) box. Then, some “typical quantum” features are recovered in the no-signaling framework, showing that they are common to all no-signaling theories.

Certifying Randomness

Nonlocality certifies that the outputs of a measurement did not pre-exist, which in particular means that they were unpredictable or random. In other words, nonlocality certifies randomness in a device-independent way. This chapter introduces the main tools for the study and quantification of randomness: process randomness, the need for a predictor or adversary, and guessing probability. Examples are then explicitly worked out, and a review of more advanced results is provided.

Device-Independent Self-Testing

This chapter is devoted to deviice-independent self-testing. This refers to the fact that some extremal behaviors in the quantum set can actually be realised only with specific measurements on a specific shared entangled state. The two main examples, namely the maximal value of CHSH and the Mayers-Yao test, are described in detail before the abstract definition of self-testing is given.

Review of Bipartite Bell Scenarios

This chapter reviews some of the Bell scenarios that go beyond CHSH, studied both with the tools of nonlocality and with quantum theory. In particular, we present the I3322 inequality, the chained inequalities, the family of CGLMP inequalities, the classic Hardy’s test and the Magic Square test.

Bell Nonlocality in Quantum Theory

All the examples of nonlocality that we know are accurately described within quantum theory. This chapter presents the study of CHSH and some systematic results. In particular, we learn that separable states can only give rise to local behaviors, while pure entangled states are nonlocal resources. The case of mixed entangled states is then introduced.

First Encounter with Bell Nonlocality

This chapter is a self-contained introduction to Bell nonlocality. After a very quick historical motivation, the setting and the definition of nonlocality are given. Five classic examples of Bell tests are then introduced, including the CHSH and the GHZ tests. The remainder of the chapter is a critical reflection that contains two topics: a discussion of possible loopholes that should be avoided, and an elementary presentation of the main interpretations given to the phenomenon of nonlocality.

Epilogue

We first stress that nonlocality forces changes on our view of the world. One can adopt the orthodox view that one should not even attempt to describe individual events; or one can opt for a properly-speaking nonlocal model, with very unpleasant features like influences propagating at infinite speed in our space-time; or one can go all the way to some form of determinism. Then, the hope is formulated that the field of nonlocality, having reached maturity, may serve as starting point for future discoveries.

Multipartite Bell Nonlocality

This chapter discusses nonlocality for scenarios with more than two players. The definition of locality is the obvious generalisation of the bipartite case, but now there are many more ways in which locality can be violated. First, the main families of inequalities are presented. Then, the definition and demonstration of genuine multipartite nonlocality is discussed.

The Set of Quantum Behaviors

Part II is devoted to the applied side of nonlocality: device-independent certification of quantumness. After an introduction to this idea, the first chapter deals with the characterisation of the set of quantum behaviors. Since this set is not easily parametrised, in practice one often works with outer approximations, membership of which can be cast as a semi-definite program.

Signaling and Measurement Dependence

This last chapter studies the relaxation of the conditions of no-signaling and measurement independence. For signaling models, after stressing the need for fine-tuning, it is shown that the speed of the hypothetical influence cannot be finite. For measurement dependence, the tolerable amount is discussed, and the related notion of randomness amplification is introduced.