e tr(pU). Actually, this result enormously simplifies the perspective on quantum measurements gained from Chapter 17. It will also be used to analyze the tricky issue of composite quantum systems, as described later. In Chapter 17, we have also established that for any operator U having a diagonal matrix with nonnegative coefficients, it is possible to associate an operator U log U with diagonal coefficients (U log U)ii = Uii log Uii. In the case U = P, we obtain the matrix: pilogpi (21.7) Pn log Pn From the above definition, the trace of p log P, with a minus sign, is given by: S(p) = -tr(plogp) = Pilog pr. (21.8) A comparison between Eqs. (21.8) and (21.1) shows that S(p) = -tr(p log p) is strictly analogous to Shannon's classical entropy, H(X), with X = {i}i=1...n being a random- event source characterized by the probability distribution {pi}. The key difference between S(p) and H(X) is that, in the former, the source is a quantum system, char- acterized not by a probability distribution, like the latter, but by a density operator p. We shall refer to S(p) as von Neumann entropy. The von Neumann (VN) entropy, thus, represents a quantum measure of the information contents in the quantum system or state under consideration, referred to as a quantum source P. Since the VN entropy measure is always equivalent to that of a classical source, H(X), it is nonnegative, or S(p) > 0 always applies. In Chapter 17, it was shown that the trace of any operator is independent of the basis representation. Thus, the density operator transformation under change of basis, i.e., p -> P = TpT+ where T is a unitary operator, is of no effect on the VN entropy. Formally: S(p)=S(TpT*)=S(p) (21.9) 21.1 Von Neumann entropy This result shows that quantum information, as defined by the VN entropy, represents an incompressible feature in quantum systems, as is also the case for classical information in random-event sources, as defined by Shannon entropy. What is the quantum infor- mation contained in a qubit? The answer is straightforward. 