This chapter presents the surprising mathematical result that classical systems can indeed have entanglement. However, the degree to which they can be entangled is strictly limited, while quantum systems have no limit to their amount of entanglement.
Entanglement is one of most peculiar characteristic of Quantum Mechanics (probably the other most mentioned is the particle tunneling through a barrier) because it entails correlations that supersedes classical correlations. It is conceived as the genuine quantum characteristic by many researchers and quite recently it has emerged as a physical resource to produce non-classical task, like quantum teleportation, quantum cryptography and quantum information. Entanglement could be distributed (efficient quantum communication is equivalent to efficient entanglement distribution (Plenio & Virmani, 2007)), concentrated (given an amount of entanglement, equal to the full content of entanglement in n pairs of identical entangled pure state systems, it is possible, using local operations on each system, to concentrate the total amount of entanglement into a smaller number m, m < n, of maximally pure entangled state system (Bennett, et al.; 1996)) and used to perform many quantum information tasks useful to overcome technical restrictions present on classical communication. In this chapter, we review the entanglement of quantum mechanical systems. First, in Section 2, we give a characterization of entanglement in terms of its special features as resource and its mathematical structure. Then, in section 3, we review some of the Bell inequalities. In Sections 4 and 5, we review some of the most important entanglement measures published for two and three entangled systems. After that, in Section 6, we give a characterization of quantum gates and operators by its entanglement power, i. e. the amount of entanglement that they can produce when acting on an arbitrary state. Finally, in Section 7, we briefly review the experimental detection of entanglement.
To dedicate a chapter to atoms and molecules in a book on quantum fluids may seem surprising at first glance. Indeed, atoms and molecules are both composite systems, possibly massive, and the first intuition is to follow the analysis of Chapter 1 , where we have classically hierarchized phases of matter according to pressure, temperature and volume. However, when looking in more detail at a standard phase diagram (see for example the case of CO 2 or water in Figure 1.3 ), one soon realizes that it exhibits many other possible interesting paths. In particular, the low pressure/low temperature limit provides an interesting region to explore and actually provides a different scenario. Indeed, in this limit, the single possible phase transition is between gas and solid. But mind that it corresponds to a very low pressure/temperature domain. There are in particular no thermal effects possibly blurring the behavior of the system. We have already seen that boson and fermion gases in their degenerate limits at zero temperature exhibit remarkable features. this particular domain of vanishing temperatures. Both atoms and molecules, depending on their total spin, can be either fermions or bosons.
All physical systems register bits of information, whether it be an atom, an electrical current, the location of a billiard ball, or a switch. Information can be classical, quantum, or a hybrid of both, depending on the system. For example, an atom or an electron or a superconducting system can register quantum information because the quantum theory applies to each of these systems, but we can safely argue that the location of a billiard ball registers classical information only. These atoms or electrons or superconducting systems can also register classical bits because it is always possible for a quantum system to register classical bits. The term information , in the context of information theory, has a precise meaning that is somewhat different from our prior “everyday” experience with it. Recall that the notion of the physical bit refers to the physical representation of a bit, and the information bit is a measure of how much we learn from the outcome of a random experiment. Perhaps the word “surprise” better captures the notion of information as it applies in the context of information theory. This chapter begins our formal study of classical information. Recall that Chapter 2 reviewed some of the major operational tasks in classical information theory. Here, our approach is somewhat different because our aim is to provide an intuitive understanding of information measures, in terms of the parties who have access to the classical systems. We define precise mathematical formulas that measure the amount of information encoded in a single physical system or in multiple physical systems. The advantage of developing this theory is that we can study information in its own right without having to consider the details of the physical system that registers it. We first introduce the entropy in Section 10.1 as the expected surprise of a random variable. We extend this basic notion of entropy to develop other measures of information in Sections 10.2–10.6 that prove useful as intuitive informational measures, but also, and perhaps more importantly, these measures are the answers to operational tasks that one might wish to perform using noisy resources. While introducing these quantities, we discuss and prove several mathematical results concerning them that are important tools for the practicing information theorist.
Quantum illumination is one of the most basic ways to realize the quantum radar for detection of low-reflectivity targets embedded in a thermal-noise bath. It preserves a strong advantage over the classical counterparts even though its fragile quantum entanglement is completely destroyed by a large amount of noise and losses. In this paper, we describe and analyze a discrete-variable quantum illumination system with ancillary degrees of freedom of entangled photon pairs. Comparing the error probability bound, i.e., the quantum Chernoff bound, we identify that this system indeed offers higher accuracy than the single-photon illumination and quantum illumination with single degree of freedom of entangled photon pairs. Furthermore, the utilization of ancillary degrees of freedom exponentially enhances signal-to-noise ratio over quantum illumination without ancillary degrees of freedom. It means that this system reduces the necessary amount of signal photons to accurately determine whether the object is present or absent in the target region, which is absolutely crucial to the stealthiness of the radar system.