In the real world, nothing is perfect and experiments have defects. EPR-based cryptography is not an exception, and one must consider the fact that detectors are not perfect, that transmission lines have losses, to mention only some problems. The paranoiac cryptography game then consists of considering that any loss could be, in fact, some information stolen by Eve. It is possible, nevertheless, to generate a perfectly safe pair of keys by a process known as key distillation, which allows Alice and Bob to combine several data which might be unsafe to make one absolutely safe data. The distillation process can be made by communication on a public channel. The drawback is obviously that the length of the key obtained at the end of the distillation process is much shorter than the number of sent pairs, so the rate of fabrication of the key can become desperately slow. Losses is nowadays a major issue, either in the Ekert method or in the BB84 method. With the most transparent optical fibers, it is difficult to obtain a sufficient rate of key distribution at a distance passing 100 kilometers. What are the solutions then to implement QKD at the scale of a country or a continent? Is it possible to use the same kind of repeaters as in classical communication with optical fibers? With classical pulses containing many photons, the pulses attenuated by propagation are still large enough to be re-amplified before a new leg of fibers. But with single photons, the situation is totally different since attenuation means that many one-photon pulses just disappear and cannot be resuscitated. A mundane solution is to have trusted nodes, that is to say, places where the key is explicitly expressed in a classical form and then re-emitted over a new leg with a fully sure quantum cryptography method. It is obviously very demanding regarding infrastructure. Moreover, from a fundamental point of view, it has the inconvenience of multiplying intermediate copies of the key, that is to say, possibilities for classical spying to happen. So the trusted nodes must be guarded by trusted security forces. Such a 2,000 kilometers semi quantum line between Beijing and Shanghai using 32 trusted nodes, has been inaugurated in 2017. An ideal method for long-distance QKD would be to dispose of quantum repeaters, that is to say, systems able to rejuvenate the quantum signal. But what does it mean in the quantum world when we use single photons? You may remember from quantum optics one, that the no-cloning theorem prevents one to create several copies of a photon with the same quantum state as the original. I will come back to this important theorem later in the lesson of today. Fortunately there is, at least in theory, a solution to extend the range of propagation of a quantum state. Its name is quantum teleportation. In quantum teleportation, a photon nu two enters an extended device of size L_T, which has as the other extremity a photon nu three. Teleportation consists of transferring the state phi of nu two to photon nu three. It is described by a unitary operation that transforms the initial product state, phi of nu two times the initial state i of nu three, into the product of an irrelevant final state of nu two by a state of nu three, which is now identical to the initial state of nu two. The state of nu two has thus been sent at a distance L_T without any loss. And since the initial photon state has been modified, the no-cloning theorem is not violated. If we had a quantum repeater of the kind just described, it would be possible to extend the distance between two entangled photons. To understand it, take nu two as a photon belonging to a pair nu one, nu two of entangled photons, separated by a distance L_EPR. Let nu two enter the repeater and be submitted to the transformation T. Before the transformation, the three photons nu one, nu two, nu three, are in the state psi of nu one, nu two, times the initial state of nu three. Applying the transformation, T, to each component, one obtains f of nu two times a state psi of nu one, nu three, identical to the initial state psi of nu one, nu two, that is to say, an EPR state. The distance between nu one and nu three is larger than the initial distance between nu one and nu two by the additional quantity L_T, that is to say, the distance of the teleportation. Quantum teleportation is one of the fascinating applications of entanglement and a key technology in many quantum communication protocols. You deserve to learn what is quantum teleportation. But in order to understand it, you must first make a detour to the so-called Bell states, an important notion in many quantum information protocols.