ing associated probabilities pi (corresponding to the discrete probability distribution p), is characterized by an entropy H(X), defined by (21.1) Quantum information theory with the continuity property x logx = 0 in the limit x 0 and the logarithm conven- tionally being in base two. Such entropy represents the average measurement of the source's information contents log(1/pi) associated with each event. As we have seen in Chapter 4, the entropy is maximum when all the source events are equiprobable, namely when pi = 1/n, which yields Hmax = log N. Consider now a quantum system. This system may exist in a quantum state 1/s ), which we shall assume here represents a statistical mixture of pure states |xi). Such pure states, which cannot be defined by any other mixtures of pure states, are orthogonal to each other and have unit length, such that (xilxj) = Sij. The set of pure states {|xi}} = {|x1), |x2> , |xn)}, thus, defines an orthonormal basis for the n-dimensional space Vn of all possible quantum states /v/s defining the system. Consistently, any state 1/>/ of Vn accepts a unique decomposition of the form (21.2) where. (i=1 n) are complex coordinates. We may choose to represent the quantum system with a state /v/ of unit length, i.e., = 1, in which case the real number pi = |xil2, represents the probability of finding the state 1/s/ in the pure state /xi). In the quantum world, the system in the quantum state 1/4/, thus, plays the role of a "random event" source and, naturally, the concepts of "information" and "entropy" may be associated with such a system. To establish such a connection, we need to use the concept of density operator (or density matrix), which was introduced in Chapter 17. As we have learnt, the density operator or matrix p is an alternative way to define a quantum system in a given state Formally, (21.3) where /xi) (Xil is the projector (or measurement) operator on the basis state /xi). It is clear that plxi)=pilxi), = which shows that /xi) is an eigenstate of p with associated eigenvalue Pi