ximum of logn for systems of dimension n made of a uniform superposition of states (S(p) = logn). Finally, the entropy is invariant under basis transformation, i.e., S(TpT+) = S(p), where T is a unitary matrix, which illustrates that quantum informa- tion is an incompressible feature in quantum systems. I shall now describe more properties of VN entropy. Relative entropy Our starting point in the investigation is the definition of relative entropy. This notion is the quantum equivalent to the classical relative entropy, or Kullback-Leibler distance, described in Chapter 5. Assume two quantum systems with density operators p and o. The relative entropy between these two systems, noted S(plo) is given by: S(pllo)=t(plogp)-tr(plogo) (21.19) For any operator, the kernel is the space described by the set of eigenvectors having zero for eigenvalues, and the support is the space described by the set of eigenvectors with nonzero eigenvalues (the union of kernel and support being the complete eigenvector space). In the definition in Eq. (21.17), the term tr(p log o) is finite only in the case where the support of p does not intersect with the kernel of o, or is fully included in the support of o (meaning (plogo)ii 0 for all diagonal matrix elements). In the contrary case, we have -tr(p logo) = +00, and the relative entropy is infinite. As a key property of relative entropy, we have S(plo)z0, (21.20) where the equality stands uniquely for the case p=0. Such a property is referred to as Klein's inequality. The basic proof of Klein's inequality is provided in Appendix U. The consequence is that for two quantum systems with density operators p and o, we have S(p) = tr(p log p) tr(p logo), a property that we shall use later when considering additivity rules for entropy. Composite system in pure state Consider, next, a composite system made of two subsystems A, B in states (VA), /VB), and assume that it is in a pure joint state, namely (v) = VA)/UB). Call PA, PB the density operators of the subsystems A, B. Then we have the pro