In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
Within the framework of Information Theory, the existence of correlations between two random variables means that we can obtain information about one of them, just by measuring or observing the other random variable. In certain cases, this kind of relationship allows obtaining information about a variable even when the other is separated by a very large distance, that is, the process of obtaining information is non-local, an example (if not the only) is the quantum entanglement. These features of correlations make it interesting and important to study, classify and quantify them. The correlations are classified into classical correlations and quantum correlations, in addition they are quantified through the mutual information. Here we will present a natural way to define classical mutual information and then we will generalize it to the quantum case. Furthermore, every term in the definitions of mutual information will be interpreted using the concepts of classical and quantum entropy.
In this paper, two conjectures which were proposed in [Phys. Rev. A \textbf{82}, 052122(2010)] on the correlations in a bipartite state
ρAB are studied. If the mutual information $I\Pa{\rho^{AB}}$ between two quantum systems
A and
B before any measurement is considered as the total amount of correlations in the state
ρAB, then it can be separated into two parts: classical correlations and quantum correlations. The so-called classical correlations $C\Pa{\rho^{AB}}$ in the state
ρAB, defined by the maximizing mutual information between two quantum systems
A and
B after von Neumann measurements on system
B, we show that it is upper bounded by the von Neumann entropies of both subsystems
A and
B, this answered the conjecture on the classical correlation. If the quantum correlations $Q\Pa{\rho^{AB}}$ in the state
ρAB is defined by $Q\Pa{\rho^{AB}}= I\Pa{\rho^{AB}} - C\Pa{\rho^{AB}}$, we show also that it is upper bounded by the von Neumann entropy of subsystem
B. It is also obtained that $Q\Pa{\rho^{AB}}$ is upper bounded by the von Neumann entropy of subsystem
A for a class of states.
In classical information theory, the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov–Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes–Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy were proposed [10,[Formula: see text]3,[Formula: see text]8,[Formula: see text]39,[Formula: see text]15,[Formula: see text]44,[Formula: see text]9,[Formula: see text]27,[Formula: see text]28,[Formula: see text]2,[Formula: see text]19,[Formula: see text]45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S-mixing entropy by Ohya [26,[Formula: see text]27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S-mixing entropy. In this paper, based on research into the dynamical entropy and mean mutual entropy of quantum systems, this study investigates the characteristics of quantum channels that exhibit entanglement, a property unique to quantum systems, and investigates the behaviour of quantum entropy for compound quantum channels by changing the perspective from information transmission between input and output to a new perspective of information transmission from the initial state to the final state.