Quantum communication is the art of transferring a quantum state from one place to another. Traditionally, the sender is named Alice and the receiver Bob. Th e basic motivation is that quantum states code quantum information -called qubits in the case of two-dimensional Hilbert spaces -and that quantum information allows tasks to be performed that could only be achieved far less effi ciently, if at all, using classical information. Th e best known example is quantum key distribution (QKD) 1-3 . In fact, there is another motivation, at least equally important to most physicists, namely the close connection between quantum communication and quantum non-locality 4,5 , as illustrated by the fascinating process of quantum teleportation 6 .
Quantum-communication theory is a broad fi eld, including for example, communication complexity 7 and quantum bit-string commitment 8 . In this review we restrict ourselves to its most promising application, QKD, both point to point and in futuristic networks.
Th ere are several ways to realize quantum communication. We list them below from the simplest to the more involved. As 'fl ying qubits' are naturally realized by photons, we oft en use 'photon' to mean 'quantum system' , although in principle, any other quantum system could do the job.
Th e basic procedure is as follows. Alice encodes the state she wants to communicate into a quantum system and sends it to Bob. Entanglement is exploited to prepare the desired quantum state at a distance. Th e quantum state is then teleported from Alice to Bob and the entanglement is also teleported -entanglement swapping.
In this review, a more intuitive perspective of quantum communication, overlooking this complexity, will be considered fi rst, starting from the basic ingredient, namely entanglement and its non-locality, continuing with weak-laser-pulse QKD and its security, before discussing quantum teleportation. Finally, a review of quantum relays as well as repeaters that require quantum memories will be given. Future challenges will be underlined throughout.
ENTANGLEMENT AND NON-LOCALITY
Entanglement is the essence of quantum physics. To understand this statement already stressed by SchrΓΆdinger 9 in 1935, it is worth presenting
Quantum communication
it in modern terms inspired by quantum-information theory. In science in general, all experimental evidence takes the form of conditional probabilities: if observer A i performs the experiment labelled x i , she observes a i and in general the probability for all of the possible results is written P(a 1 β¦a n |x 1 β¦x n ). Such conditional probabilities are oft en called correlations. For simplicity, we restrict the discussion here to the bipartite case, denoting their correlation P (a,b|x,y).
Th e correlations P(a,b|x,y) carry a lot of structure. Apart from being non-negative and normalized, the local marginals are independent of the experiment performed by independent observers: β a P(a,b|x,y) = P(b|y), is independent of the experiment x performed by Alice. As a trivial example of independent observers, imagine two physicists performing diff erent experiments in labs in distant countries, in which case the independence of the marginals is obvious. Th ere is, however, another more interesting situation. Suppose the two parties perform similar experiments, but at two space-like separated locations, thus preventing any communication, as is the case in Fig. 1. It is therefore natural to assume that the local probabilities depend only on the local state of aff airs and, as the local state of aff airs may be unknown, one merely denotes them by a generic symbol, Ξ». Note that the local state of aff airs at Alice's site and at Bob's site may still be correlated. Th is is why computer scientists call Ξ» shared randomness. Given the local state of aff airs, the correlations factorize to local correlations, P(a,b|x,y,Ξ») = P(a|x,Ξ») β’ P(b|y,Ξ»), which necessarily satisfy some (infi nite) set of inequalities, known as Bell inequalities 5,10 . Let us emphasize that there is no need to assume predetermined values to derive Bell inequalities, it suffi ces to assume that the probabilities of results of local experiments depend only on local variables.
Almost all correlations between independent observers known in science are local. Th e only exceptions are some correlations predicted by quantum physics, when the two observers perform measurements on two (or more) entangled systems. Th is implies that in some cases, a quantum experiment performed at two distant locations can't be completely described by the local state of aff airs 5 , a very surprising prediction of quantum physics indeed.
Einstein, among others, was so surprised by this that he concluded that it 'proves' the incompleteness of quantum mechanics 11 . Following Bohr's reply to the famous EPR (Einstein, Podolsky and Rosen) paper, the debate became philosophical. John Bell resolved this with the introduction of the experimental question of Bell inequalities 5,10,12 and remarkably, by 1991 it had become applied physics 2 . Indeed, it was realized that the Quantum communication, and indeed quantum information in general, has changed the way we think about quantum physics. In 1984 and 1991, the fi rst protocol for quantum cryptography and the fi rst application of quantum non-locality, respectively, attracted interest from a diverse fi eld of researchers in theoretical and experimental physics, mathematics and computer science. Since then we have seen a fundamental shift in how we understand information when it is encoded in quantum systems.
We review the current state of research and future directions in this fi eld of science with special emphasis on quantum key distribution and quantum networks.