Quantum Information Theory

Abstract

We have already come across the Shannon and von Neumann entropies in the chapter on entanglement. The Shannon entropy assesses the information content of a classical probability distribution while the von Neumann entropy does the same for its quantum counterpart, the quantum state (density matrix) of a quantum system. We have also come across the trace distance and the fidelity in the chapter on quantum states. These quantities proved very useful when we wanted to compare different states of a quantum system. However, these distance measures were introduced ad hoc, without introducing their classical counterpart first and then showing that these are the proper, or at least the obvious, quantum generalizations. Therefore, in the first two sections of this chapter, we will introduce these quantities in a more systematic manner. Then, we want to use the entropic quantities to say something about quantum communication, in particular, sending classical messages using quantum systems. In order to do so, we will need to define Shannon and von Neumann entropies for two random variables, in the classical case, and two systems, in the quantum case, which we do in the third section. It is also in order here to introduce consistent notation: entropic quantities related to classical probability distributions will be denoted by H and then further specified by their arguments, and their quantum counterparts, entropic quantities related to quantum states (density matrices), will be denoted by S and, again, further specified by their arguments. In the remaining sections, we will apply these quantities to studying quantum signatures of communication, establishing some important bounds and discussing an illustrative example.