Quantum Relative Entropy - Motivation

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...p_n}, but somehow we mistakenly assumed it to be Q = {q₁...q_n}. 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

1. In the definitions above, the convention that 0·log 0 = 0 is assumed, since lim_{x → 0} x log x = 0. Intuitively, one would expect that an event of zero probability to contribute nothing towards entropy.
2. 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.