 the two reals defined by Pn = anPn-1 + Pn-2 an=anqn-1+ qn-2 for n 2, and for n = 0, 1, satisfy the relation qnPn-1-Pnqn-1=(-1)". - Quantum information theory This chapter sets the basis of quantum information theory (QIT). The central purpose of QIT is to qualify the transmission of either classical or quantum information over quantum channels. The starting point of the QIT description is von Neumann entropy, S(p), which represents the quantum counterpart of Shannon's classical entropy, H(X). Such a definition rests on that of the density operator (or density matrix) of a quantum system, p, which plays a role similar to that of the random-events source X in Shannon's theory. As we shall see, there also exists an elegant and one-to-one correspondence between the quantum and classical definitions of the entropy variants relative entropy, joint entropy, conditional entropy, and mutual information. But such a similarity is only apparent. Indeed, one becomes rapidly convinced from a systematic analysis of the entropy's additivity rules that fundamental differences separate the two worlds. The classical notion of information correlation between two event sources for quantum states shall be referred to as quantum entanglement. We then define a quantum communication channel, which encodes and decodes classical information into or from quantum states. The analysis shows that the mutual information H(X; Y) between originator and recipient in this communication channel cannot exceed a quantity X, called the Holevo bound, which itself satisfies X < H(X), where H(X) is the entropy of the originator's classical information source. It is shown that the Holevo bound is maximal (x = H(X)) when the set of density operators used for coding the message have orthogonal supports or eigenspaces. Von Neumann entropy The concept of source entropy, as introduced in Shannon's classical information theory, has been extensively described and developed in Chapters 4 to 6. To recall, a source X of n random events labeled i (I = 1...n) and hav