The astute reader might have formed the impression that quantum in formation science is a rather qualitative discipline because we have not, as yet, explained how to quantify quantum information. There are three good reasons for leaving this important question until the final chapter. Firstly, quantum information theory is technically demanding and to treat it at an earlier stage might have suggested that our subject was more complicated than it is. Secondly, there is the fact that many of the ideas in the field, such as teleportation and quantum circuits, are unfamiliar and it was important to present these as simply as possible. Finally, and most importantly, the theory of quantum information is not yet fully developed. It has not yet reached, in particular, the level of completeness of its classical counterpart. For this reason we can answer only some of the many questions we would like a quantum theory of information to address. Having said this, we can say that however, there are beautiful and useful mathematical results and it seems certain that these will continue to form an important part of the theory as it develops. We noted in the introduction to Chapter 1 that ‘quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed’. A presentation of at least the beginnings of a quantitative theory is the objective of this final chapter. The entropy or information derived from a given probability distribution is, as we have seen, a convenient measure of the uncertainty associated with the distribution. If many of the probabilities are large, so that many of the possible events are comparably likely, then the entropy will be large. If one probability is close to unity, however, then the entropy will be small. It is convenient to introduce entropy in quantum mechanics as a measure of the uncertainty, or lack of knowledge, of the form of the state vector. If we know that our system is in a particular pure state then the associated uncertainty or entropy should be zero. For mixed states, however, it will take a non-zero value.
Quantum Information Theory and the Foundations of Quantum Mechanics
