Hi, welcome to a new lesson of our Quantum Optics course Season 2. Today, you will encounter for the first time in this course, the notion of entanglement, which you have probably heard of. If you wanted to know more, but you were afraid to ask you are at the right place to get some answers. Entanglement is a notion which is built-in in the basic formalism of quantum mechanics, as soon as you want to describe a system made of more than one particle. Entangled states play a role in many different branches of physics. For instance in atomic physics, you need to write entangled states to describe the two electrons of the helium atom. In condensed matter physics, it has been known for a long time that the many electrons of a piece of matter should be represented as a big entangled state, even without taking into account interactions between them. When interactions are taken into account, an exact description of the electrons is beyond the possibility of the standard formalism. This is known as the many-body problem. For decades however, entanglement was considered an ordinary feature of quantum mechanics, not more surprising than wave-particle duality, although presenting major difficulties for theorists. It is only in the last decades of the 20th century that physicist progressively realized that entanglement is definitely different from any quantum feature associated with wave particle duality. An emblematic example of this late awareness can be found in two citations of Richard Feynman, separated by two decades. In 1960, in his famous lectures on physics, which every physicist should have in his or her personal library, he presents wave-particle duality for a single particle. He then claims that it is a big mystery and that it is the only mystery. He then comments that there is nothing more in the Einstein-Podolsky-Rosen situation, which is in fact an emblematic example of entanglement, as you will learn today. But two decades later, Feynman published a paper where he acknowledged that he had missed another mystery. Then he went on and explained how the violation of inequalities points to another mysterious feature. Although he does not name them, these inequalities are Bell's inequalities, and the other mystery is entanglement. An important part of Feynman's paper is the vision that when you have a new concept, you can use it for something new, and he introduces basic ideas of quantum computing. The second quantum revolution is based on the ability to manipulate and observe single quantum objects, and to control entanglement between these individual quantum objects. In this lesson, you will get fully acquainted with the example of polarization entanglement in pairs of photons, a system emblematic of modern quantum optics. Pairs of entangled photons have been the privileged system to test Bell's inequalities and confirm the extraordinary nature of entanglement. They are also a basic resource for quantum information. I invite you, then, to embark into the magic and mysterious world of entanglement, not only in theoretical discussions but also in experiments. Entanglement in its most general sense is a complex subject not fully clarified, but we will restrict our lesson to the simple case of two entangled photons. It will allow you to discover the most fundamental and surprising properties of entanglement and is at the root of remarkable applications as you will see in a future lesson. In Section 1, I will recall that a polarized single photon wave packet is an almost ideal two level system, an ideal quantum bit. In Section 2, you will learn about the emblematic example of entanglement two photons entangled in polarization. Using standard methods of quantum optics, you will find that polarization measurements are strongly correlated. These strong correlations are of the kind discovered by Albert Einstein, Boris Podolsky, and Nathan Rosen in 1935. In Section 3, we will discuss the possibility to interpret these correlations in the spirit of Einstein's vision of the world. One must admit that this point of view is extremely reasonable in spite of the opposition of Niels Bohr. In Section 4, you will see that adopting Einstein's worldview, called local realism, puts constraints on the correlations. These constraints are expressed in inequalities discovered by John Stewart Bell in 1964. Surprisingly, it turns out that correlations predicted by the quantum formalism for an EPR state, as calculated in Section 2, violate Bell's inequalities, that is to say, are incompatible with Einstein's local realist vision of the world. In Section 5, I will rapidly present experiments that have been carried out to settle the debate between Einstein and Bohr. The result is unambiguous. Experiments confirm the predictions of quantum mechanics, and violate Bell's inequalities, thus rejecting the local realist worldview. In conclusion, I will emphasize that beyond the conceptual debate, the violation of Bell's inequalities has drawn attention onto extraordinary properties of entanglement: quantum non-locality and quantum holism. As you will learn in a future lesson, these two concepts are at the root of the development of quantum technologies in the framework of the second quantum revolution.