Quantum relative entropy
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