Chapter 2 clarifies the connection between Bananaworld correlations and quantum correlations with an introduction to the simplest quantum systems, qubits, using polarized photons – particles of light – as the standard example. A “bit” is the basic unit of classical (Shannon) information, but the term is also used to refer to an elementary classical system that can be in one of two states. The corresponding elementary quantum system is referred to as a “qubit,” short for “quantum bit.” A photon is a qubit with respect to its state of polarization. The Born rule specifies the probability that a photon in a certain initial polarization state, prepared by passing the photon through a polarizing filter, will be detected by a photon detector after being directed to pass through a second polarizer called an “analyzer.” A pair of qubits can be in an “entangled” state. Two separated “maximally entangled” photons exhibit nonclassical correlations for appropriate polarization measurements analogous to Bananaworld correlations.
Chapter 2 clarifies the connection between Bananaworld correlations and quantum correlations with an introduction to the simplest quantum systems, qubits, using polarized photons – particles of light – as the standard example. A “bit” is the basic unit of classical (Shannon) information, but the term is also used to refer to an elementary classical system that can be in one of two states. The corresponding elementary quantum system is referred to as a “qubit,” short for “quantum bit.” A photon is a qubit with respect to its state of polarization. The Born rule specifies the probability that a photon in a certain initial polarization state, prepared by passing the photon through a polarizing filter, will be detected by a photon detector after being directed to pass through a second polarizer called an “analyzer.” A pair of qubits can be in an “entangled” state. Two separated “maximally entangled” photons exhibit nonclassical correlations for appropriate polarization measurements analogous to Bananaworld correlations.
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.
In this chapter we build up the basic concepts of quantum mechanics using two simple examples, following a heuristic approach which is more inductive than deductive. We start with a familiar phenomenon, that of the polarization of light, which will allow us to introduce the necessary mathematical formalism. We show that the description of polarization leads naturally to the need for a two-dimensional complex vector space, and we establish the correspondence between a polarization state and a vector in this space, referred to as the space of polarization states. We then move on to the quantum description of photon polarization and illustrate the construction of probability amplitudes as scalar products in this space. The second example will be that of spin 1/2, where the space of states is again two-dimensional. We construct the most general states of spin 1/2 using rotational invariance. Finally, we introduce dynamics, which allows us to follow the time evolution of a state vector. The analogy with the polarization of light will serve as a guide to constructing the quantum theory of photon polarization, but no such classical analog is available for constructing the quantum theory for spin 1/2. In this case the quantum theory will be constructed without reference to any classical theory, using an assumption about the dimension of the space of states and symmetry principles. The polarization of light and photon polarization The polarization of an electromagnetic wave The polarization of light or, more generally, of an electromagnetic wave, is a familiar phenomenon related to the vector nature of the electromagnetic field. Let us consider a plane wave of monochromatic light of frequency ω propagating in the positive z direction.
Abstract Entanglement and polarization of pure, two-qubit systems are known to be constrained by the relationship C 2 + P 2 = 1 . Here, C stands for concurrence, a standard measure of entanglement, and P refers to the degree of polarization of either of the two, single-qubit subsystems. The above constraint may be understood as reflecting that a system cannot be in a pure state, if it is entangled with another one. In order to explore the connection between entanglement and polarization beyond two-qubit systems, we addressed states in which at least one of the parties has dimensionality greater than two. The case of pure, tripartite states offered itself as a timely candidate. We focussed on three qubits and studied the connection between one qubit’s polarization and its entanglement with the other two taken together. We derived new constraints, akin to the above one, and submitted them to experimental tests using single photons entangled in polarization. The latter provided two qubits, each one attached to one photon. The third qubit was chosen to be the path (momentum) of one of the photons. Our experimental results confirmed the validity of the new constraints. Our theoretical results hold also for classical light. Possible experimental tests could be done with so-called structured light beams.
The simplest quantum system is the physical quantum bit or qubit . The qubit is a two-level quantum system—example qubit systems are the spin of an electron, the polarization of a photon, or a two-level atom with a ground state and an excited state. We do not worry too much about physical implementations in this chapter, but instead focus on the mathematical postulates of the quantum theory and operations that we can perform on qubits. From qubits we progress to a study of physical qudits . Qudits are quantum systems that have d levels and are an important generalization of qubits. Again, we do not discuss physical realizations of qudits. Noise can affect quantum systems, and we must understand methods of modeling noise in the quantum theory because our ultimate aim is to construct schemes for protecting quantum systems against the detrimental effects of noise. In Chapter 1, we remarked on the different types of noise that occur in nature. The first, and perhaps more easily comprehensible type of noise, is that which is due to our lack of information about a given scenario. We observe this type of noise in a casino, with every shuffle of cards or toss of dice. These events are random, and the random variables of probability theory model them because the outcomes are unpredictable. This noise is the same as that in all classical information-processing systems. On the other hand, the quantum theory features a fundamentally different type of noise. Quantum noise is inherent in nature and is not due to our lack of information, but is due rather to nature itself. An example of this type of noise is the “Heisenberg noise” that results from the uncertainty principle. If we know the momentum of a given particle from performing a precise measurement of it, then we know absolutely nothing about its position—a measurement of its position gives a random result. Similarly, if we know the rectilinear polarization of a photon by precisely measuring it, then a future measurement of its diagonal polarization will give a random result. It is important to keep the distinction clear between these two types of noise.
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 Neumman entropy. It is the quantum analog of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state. The von Neumann entropy gives meaning to a notion of the information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state in an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is in a quantum system. The beginning 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 a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical spatial 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.
The physical meaning of the basic quantum mechanical concepts (such as the wave function, reduction, state preparation and measurement, the projection postulate, and the uncertainty principle) is clarified using realistic experimental procedures and employing classical analogies whenever possible. Photon polarization measurement and particle coordinate and momentum measurement are considered as examples, as also are Einstein–Podolsky–Rosen correlations, Aharonov–Bohm effects, quantum teleportation, etc. Various nonclassicality criteria of quantum models, including photon antibunching and the violation of the Bell inequality, are discussed.