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This series of introductory lectures consists of two parts. The first part reviews the basic notions of quantum physics and some primitives of quantum information. The latter are notions that one must somehow be familiar with in the field: quantum cloning, teleportation and entanglement swapping, entanglement distillation, state estimation (both single shot and multi-copy), and quantum coding. The second part is devoted to a detailed introduction to the topic of quantum correlations. The evidence for failure of alternative theories is presented: the violation of Bell’s inequalities is dealt with in detail, other tests like Leggett’s inequalities and the before-before experiment are discussed. The last two lectures introduce the formalism of no-signaling probability distributions and explain the possibility of device-independent quantum information.

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.

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.

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.

Classical physics cannot guarantee the security of a procedure for distributing a cryptographic key between two parties. In a quantum world, any attempt by Eve to eavesdrop on communications between Alice and Bob can be revealed. The chapter shows how Alice and Bob can share secrets with unconditional security in Bananaworld by exploiting the “monogamous” correlations of Popescu–Rohrlich bananas, and discusses a protocol by Ekert that exploits the monogamy of entanglement in our quantum world to allow Alice and Bob to share a secret key. Finally, the chapter shows how this can be extended to a device-independent scenario, where Alice and Bob can share a secret key even if they suspect that the devices they use to prepare and measure entangled quantum states are insecure and might have been supplied by an adversary.