In quantum information theory, quantum relative entropy is a measure of distinguishability between two quantum states. It is the quantum mechanical analog of relative entropy. Motivation For simplicity, it will be assumed that all objects in the article are finite-dimensional. We first discuss the classical case. Suppose the probabilities of a finite sequence of events is given by the probability distribution P = {p1...pn}, but somehow we mistakenly assumed it to be Q = {q1...qn}. For instance, we can mistake an unfair coin for a fair one. According to this erroneous assumption, our uncertainty about the j-th event, or equivalently, the amount of information provided after observing the j-th event, is The (assumed) average uncertainty of all possible events is then On the other hand, the Shannon entropy of the probability distribution p, defined by is the real amount of uncertainty before observation. Therefore the difference between these two quantities is a measure of the distinguishability of the two probability distributions p and q. This is precisely the classical relative entropy, or Kullback–Leibler divergence: Note In the definitions above, the convention that 0·log 0 = 0 is assumed, since . Intuitively, one would expect that an event of zero probability to contribute nothing towards entropy. The relative entropy is not a metric. For example, it is not symmetric. The uncertainty discrepancy in mistaking a fair coin to be unfair is not the same as the opposite situation. Definition As with many other objects in quantum information theory, quantum relative entropy is defined by extending the classical definition from probability distributions to density matrices. Let ρ be a density matrix. The von Neumann entropy of ρ, which is the quantum mechanical analog of the Shannon entropy, is given by For two density matrices ρ and σ, the quantum relative entropy of ρ with respect to σ is defined by We see that, when the states are classically related, i.e. ρσ = σρ, the definition coincides with the classical case, in the sense that if and with and (because and commute, they are simultaneously diagonalizable), then is just the ordinary Kullback-Leibler divergence of the probability vector with respect to the probability vector . Non-finite (divergent) relative entropy In general, the support of a matrix M is the orthogonal complement of its kernel, i.e. . When considering the quantum relative entropy, we assume the convention that −s · log 0 = ∞ for