any s > 0. This leads to the definition that when This can be interpreted in the following way. Informally, the quantum relative entropy is a measure of our ability to distinguish two quantum states where larger values indicate states that are more different. Being orthogonal represents the most different quantum states can be. This is reflected by non-finite quantum relative entropy for orthogonal quantum states. Following the argument given in the Motivation section, if we erroneously assume the state has support in , this is an error impossible to recover from. However, one should be careful not to conclude that the divergence of the quantum relative entropy implies that the states and are orthogonal or even very different by other measures. Specifically, can diverge when and differ by a vanishingly small amount as measured by some norm. For example, let have the diagonal representation with for and for where is an orthonormal set. The kernel of is the space spanned by the set . Next let for a small positive number . As has support (namely the state ) in the kernel of , is divergent even though the trace norm of the difference is . This means that difference between and as measured by the trace norm is vanishingly small as even though is divergent (i.e. infinite). This property of the quantum relative entropy represents a serious shortcoming if not treated with care. Non-negativity of relative entropy Corresponding classical statement For the classical Kullback–Leibler divergence, it can be shown that and the equality holds if and only if P = Q. Colloquially, this means that the uncertainty calculated using erroneous assumptions is always greater than the real amount of uncertainty. To show the inequality, we rewrite Notice that log is a concave function. Therefore -log is convex. Applying Jensen's inequality, we obtain Jensen's inequality also states that equality holds if and only if, for all i, qi = (Σqj) pi, i.e. p = q. The result Klein's inequality states that the quantum relative entropy is non-negative in general. It is zero if and only if ρ = σ. Proof Let ρ and σ have spectral decompositions So Direct calculation gives where Pi j = |vi*wj|2. Since the matrix (Pi j)i j is a doubly stochastic matrix and -log is a convex function, the above expression is Define ri = Σjqj Pi j. Then {ri} is a probability distribution.