We cannot overstate the importance of Shannon's contribution to modern science. His introduction of the field of information theory and his solutions to its two main theorems demonstrate that his ideas on communication were far beyond the other prevailing ideas in this domain around 1948. In this chapter, our aim is to discuss Shannon's two main contributions in a descriptive fashion. The goal of this high-level discussion is to build up the intuition for the problem domain of information theory and to understand the main concepts before we delve into the analogous quantum information-theoretic ideas. We avoid going into deep technical detail in this chapter, leaving such details for later chapters where we formally prove both classical and quantum Shannon-theoretic coding theorems. We do use some mathematics from probability theory, namely, the law of large numbers. We will be delving into the technical details of this chapter's material in later chapters (specifically, Chapters 10, 12, and 13). Once you have reached later chapters that develop some more technical details, it might be helpful to turn back to this chapter to get an overall flavor for the motivation of the development. Data Compression We first discuss the problem of data compression. Those who are familiar with the Internet have used several popular data formats such as JPEG, MPEG, ZIP, GIF, etc. All of these file formats have corresponding algorithms for compressing the output of an information source.
We cannot overstate the importance of Shannon's contribution to modern science. His introduction of the field of information theory and his solutions to its two main theorems demonstrate that his ideas on communication were far beyond the other prevailing ideas in this domain around 1948. In this chapter, our aim is to discuss Shannon's two main contributions in a descriptive fashion. The goal of this high-level discussion is to build up the intuition for the problem domain of information theory and to understand the main concepts before we delve into the analogous quantum information-theoretic ideas. We avoid going into deep technical detail in this chapter, leaving such details for later chapters where we formally prove both classical and quantum Shannontheoretic coding theorems. We do use some mathematics from probability theory (namely, the law of large numbers). We will be delving into the technical details of this chapter's material in later chapters (specifically, Chapters 10, 13, and 14). Once you have reached later chapters that develop some more technical details, it might be helpful to turn back to this chapter to get an overall flavor for the motivation of the development. Data Compression We first discuss the problem of data compression. Those who are familiar with the Internet have used several popular data formats such as JPEG, MPEG, ZIP, GIF, etc. All of these file formats have corresponding algorithms for compressing the output of an information source. A first glance at the compression problem might lead one to believe that it is not possible to compress the output of the information source to an arbitrarily small size, and Shannon proved that this is the case. This result is the content of Shannon's first noiseless coding theorem. An Example of Data Compression We begin with a simple example that illustrates the concept of an information source. We then develop a scheme for coding this source so that it requires fewer bits to represent its output faithfully. Suppose that Alice is a sender and Bob is a receiver. Suppose further that a noiseless bit channel connects Alice to Bob—a noiseless bit channel is one that transmits information perfectly from sender to receiver, e.g., Bob receives “0” if Alice transmits “0” and Bob receives “1” if Alice transmits “1.”
One of the fundamental tasks in classical information theory is the compression of information. Given access to many uses of a noiseless classical channel, what is the best that a sender and receiver can make of this resource for compressed data transmission? Shannon's compression theorem demonstrates that the Shannon entropy is the fundamental limit for the compression rate in the IID setting (recall the development in Section 13.4). That is, if one compresses at a rate above the Shannon entropy, then it is possible to recover the compressed data perfectly in the asymptotic limit, and otherwise, it is not possible to do so. This theorem establishes the prominent role of the entropy in Shannon's theory of information. In the quantum world, it very well could be that one day a sender and a receiver would have many uses of a noiseless quantum channel available, and the sender could use this resource to transmit compressed quantum information. But what exactly does this mean in the quantum setting? A simple model of a quantum information source is an ensemble of quantum states { pX(x) , ∣ψ x ⟩}, i.e., the source outputs the state ∣ψ x ⟩ with probability pX(x) , and the states {∣ψ x ⟩} do not necessarily have to form an orthonormal basis. Let us suppose for the moment that the classical data x is available as well, even though this might not necessarily be the case in practice. A naive strategy for compressing this quantum information source would be to ignore the quantum states coming out, handle the classical data instead, and exploit Shannon's compression protocol from Section 13.4.