A (Boolean) bit assumes two distinct values, 0 and 1. Bits constitute the building blocks of the classical information theory founded by C. Shannon. Quantum information theory, on the other hand, is based on qubits.
In these first few chapters, our aim is to establish a firm grounding so that we can address some fundamental questions regarding information transmission over quantum channels. This area of study has become known as “quantum Shannon theory” in the broader quantum information community, in order to distinguish this topic from other areas of study in quantum information science. In this text, we will use the terms “quantum Shannon theory” and “quantum information theory” somewhat interchangeably. We will begin by briefly overviewing several fundamental aspects of the quantum theory. Our study of the quantum theory, in this chapter and future ones, will be at an abstract level, without giving preference to any particular physical system such as a spin-1/2 particle or a photon. This approach will be more beneficial for the purposes of our study, but, here and there, we will make some reference to actual physical systems to ground us in reality. You may be wondering, what is quantum Shannon theory and why do we name this area of study as such? In short, quantum Shannon theory is the study of the ultimate capability of noisy physical systems, governed by the laws of quantum mechanics, to preserve information and correlations. Quantum information theorists have chosen the name quantum Shannon theory to honor Claude Shannon, who single-handedly founded the field of classical information theory, with a groundbreaking 1948 paper (Shannon, 1948).
In these first few chapters, our aim is to establish a firm grounding so that we can address some fundamental questions regarding information transmission over quantum channels. This area of study has become known as “quantum Shannon theory” in the broader quantum information community, in order to distinguish this topic from other areas of study in quantum information science. In this text, we will use the terms “quantum Shannon theory” and “quantum information theory” somewhat interchangeably. We will begin by briefly overviewing several fundamental aspects of the quantum theory. Our study of the quantum theory, in this chapter and future ones, will be at an abstract level, without giving preference to any particular physical system such as a spin- 1/2 particle or a photon. This approach will be more beneficial for the purposes of our study, but, here and there, we will make some reference to actual physical systems to ground us in reality. You may be wondering, what is quantum Shannon theory and why do we name this area of study as such? In short, quantum Shannon theory is the study of the ultimate capability of noisy physical systems, governed by the laws of quantum mechanics, to preserve information and correlations. Quantum information theorists have chosen the name quantum Shannon theory to honor Claude Shannon, who single-handedly founded the field of classical information theory with a groundbreaking paper (Shannon, 1948). In particular, the name refers to the asymptotic theory of quantum information, which is the main topic of study in this book. Information theorists since Shannon have dubbed him the “Einstein of the information age.”1 The name quantum Shannon theory is fit to capture this area of study because we often use quantum versions of Shannon's ideas to prove some of the main theorems in quantum Shannon theory. We prefer the name “quantum Shannon theory” over such names as “quantum information science” or just “quantum information.” These other names are too broad, encompassing subjects as diverse as quantum computation, quantum algorithms, quantum complexity theory, quantum communication complexity, entanglement theory, quantum key distribution, quantum error correction, and even the experimental implementation of quantum protocols.
Classical information processing is performed using bits , which are just two-state systems, with the two states called 0 and 1. By grouping bits togetherwe can represent arbitrary pieces of information, and by manipulating these bits we can perform arbitrary computations. The corresponding basic element used in quantum information is the quantum bit, or qubit . This is simply a quantum system with two orthonormal basis states, which we shall call |0⧽ and |1⧽. There are many possible physical implementations of a qubit, such as spin states of electrons or atomic nuclei, charge states of quantum dots, atomic energy levels, vibrational states of groups of atoms, polarization states of photons, or paths in an interferometer. At this stage the physical implementation is not important: the idea of a qubit is to abstract the discussion away from physical details. Taking the standard approach of quantum information theory, we shall begin by not worrying too much about the properties of these states, or even what their energies are; we shall simply assume that they are eigenstates of the system's Hamiltonian with known eigenvalues (that is, known energies). This approach allows us to concentrate on the fundamental properties of the system, without considering all the tedious details. We can in principle perform classical information processing on our quantum system by using the two states |0⧽ and |1⧽ as our logical states 0 and 1 and proceeding in the usual fashion, giving rise to the field of reversible computing , which will be explored briefly in Part II. This, however, misses the point.
Quantum information is defined by applying the concepts of ordinary (Shannon) information theory to a quantum sample space consisting of a single framework or consistent family. A classical analogy for a spin-half particle and other arguments show that the infinite amount of information needed to specify a precise vector in its Hilbert space is not a measure of the information carried by a quantum entity with a d-dimensional Hilbert space; the latter is, instead, bounded by log2d bits (one bit per qubit). The two bits of information transmitted in dense coding are located not in one but in the correlation between two qubits, consistent with this bound. A quantum channel can be thought of as a structure or collection of frameworks, and the physical location of the information in the individual frameworks can be used to identify the location of the channel. Analysis of a quantum circuit used as a model of teleportation shows that the location of the channel depends upon which structure is employed; for ordinary teleportation it is not (contrary to Deutsch and Hayden) present in the two bits resulting from the Bell-basis measurement, but in correlations of these with a distant qubit. In neither teleportation nor dense coding does information travel backwards in time, nor is it transmitted by nonlocal (superluminal) influences. It is (tentatively) proposed that all aspects of quantum information can in principle be understood in terms of the (basically classical) behavior of information in a particular framework, along with the framework dependence of this information.
To build a general-purpose quantum computer, it is crucial for the quantum devices to implement classical boolean logic. A straightforward realization of quantum boolean logic is to use auxiliary qubits as intermediate storage. This inefficient implementation causes a large number of auxiliary qubits to be used. In this paper, we have derived a systematic way of realizing any general m-to-n bit combinational boolean logic using elementary quantum gates. Our approach transforms the m-to-n bit classical mapping into a t-bit unitary quantum operation with minimum number of auxiliary qubits, then a variation of Toffoli gate is used as the basic building block to construct the unitary operation. Finally, each of these building blocks can be decomposed into one-bit rotation and two-bit control-U gates. The efficiency of the network is taken into consideration by formulating it as a constrained set partitioning problem.