(17.39) We now have the tools to make a short hint at quantum information theory. This may also constitute a nice reward for having gone through the lengthy description of Dirac notation! Let us introduce a new operator, called U log U, where U is assumed to be diagonal with nonnegative coefficients. To calculate the matrix coefficients of U log U, we must first define log U. Assume, then, a linear operator V. which satisfies U = exp(V), which defines V = log U. Formally, the exponential operator is determined by the infinite series: (17.40) Since U is diagonal, any of the powers Vn must be diagonal. The diagonal coefficients of U are, thus, given by Uii = exp(Vii) or Vii = log(Ui). The matrix W = UV = U log U is also diagonal. It is clear that its coefficients are given by W = Uii Vii = Uii log(Ui). This result shows that the matrix W = U log U is analytically defined for any diagonal matrix U with coefficients Uii > 0.4 We conclude that the density matrix U = p, for which the coefficients are nonnegative, is an eligible candidate for the operator U log U. We, thus, have (p log P)ii = Pii log Pii and the matrix definition |x1|2 1xn12 (17.41) Finally, we find that the trace of p log p is given by the expression: (17.42) Based on our background of Shannon's information theory (Chapter 4), we can heuris- tically define an "entropy" H for the quantum state described by the density matrix p in the form (17.43) 4 By application of the property limx->(o) logx) = 0, which defines the function x logx analytically for any x 0 0. 17.2 Quantum measurements and types which, from Eq. (17.42), yields: H =-tr(plogp). (17.44) The above result quite elegantly connects the concept of Shannon's entropy to the corre- sponding "notion" in quantum information theory, which is based on the density-matrix operator. The definition H = -tr(p log p) is referred to as von Neumann's entropy, as I shall describe in Chapter 21. Anticipating a key result, the entropy corresponding to a qubit state |q) = a|0) + B|1), with p=1al2=1-1812, is given