 exercises), assume the following density operator: (21.15) In the computational basis {|0), |1)}, it is easily established that the corresponding matrix is (21.16) As recalled in Chapter 17, the eigenvalues di are given by the solutions a of the char- acteristic equation, i.e., det(p - AI) = 0. By definition, each eigenvalue corresponds to an eigenvector (2i), for which p/2i) = dildi), with {12i)} constituting an orthonormal basis. In the example, the solutions of the characteristic equation are easily found to be 2 = (3 5)/6, which, in the eigenstate-basis representation {|2i}}, yields the diagonal density matrix:2 (21.17) We can then determine the VN entropy of the associated quantum system, which is in the state now represented as 1/5/ = V21/21) + V22/22), as follows: S(p) = -tr(plogp) (21.18) 0.55004. This result concludes that this system contains quantum information amounting to about 0.55 classical bits, as measured when using the eigenstate basis {|2i}}. The connection between classical and quantum information, as measured by Shannon and von Neumann entropies, appears to be elegant and complete, almost to the extent of hiding the fundamental differences between their respective worlds. As the follow- ing sections will show, however, it would be largely erroneous to conclude from this connection that QIT brings nothing new to information theory. It is easily checked that the same density operator p is obtained by starting from the computational basis {l+>,|->} instead of {|0), |1)}. 21.2 Relative, joint, and conditional entropy, and mutual information Relative, joint, and conditional entropy, and mutual information The properties of VN entropy (henceforth to be referred to simply as "entropy") are listed in this section, along with those of related entropy variants that represent the quantum counterparts of similar definitions in classical theory. To recall the basics, the previous section showed that the VN entropy is nonneg- ative (S(p) > 0), is zero for systems in pure states (S(p) = = 0), and has a ma