Shannon's theory of information was built on the assumption that the information carriers were classical systems. Its quantum counterpart, quantum Shannon theory, explores the new possibilities that arise when the information carriers are quantum particles. Traditionally,quantum Shannon theory has focussed on scenarios where the internal state of the particles is quantum, while their trajectory in spacetime is classical. Here we propose a second level of quantisation where both the information and its propagation in spacetime is treated quantum mechanically. The framework is illustrated with a number of examples, showcasing some of the couterintuitive phenomena taking place when information travels in a superposition of paths.
Shannon's theory of information was built on the assumption that the information carriers were classical systems. Its quantum counterpart, quantum Shannon theory, explores the new possibilities that arise when the information carriers are quantum particles. Traditionally,quantum Shannon theory has focussed on scenarios where the internal state of the particles is quantum, while their trajectory in spacetime is classical. Here we propose a second level of quantisation where both the information and its propagation in spacetime is treated quantum mechanically. The framework is illustrated with a number of examples, showcasing some of the couterintuitive phenomena taking place when information travels in a superposition of paths.
Shannon's theory of information was built on the assumption that the information carriers were classical systems. Its quantum counterpart, quantum Shannon theory, explores the new possibilities arising when the information carriers are quantum systems. Traditionally, quantum Shannon theory has focused on scenarios where the internal state of the information carriers is quantum, while their trajectory is classical. Here we propose a second level of quantization where both the information and its propagation in space-time is treated quantum mechanically. The framework is illustrated with a number of examples, showcasing some of the counterintuitive phenomena taking place when information travels simultaneously through multiple transmission lines.
Shannon's theory of information laid the groundwork for the rapid developments in information and communications technologies over the last century. Yet, it assumed that information carriers were described by the laws of classical physics, whilst at the most fundamental level, nature obeys the laws of quantum physics. Quantum Shannon theory, which describes information carriers as quantum states, has led to a new era of possibilities, such as perfectly secure cryptography without pre-established keys. Yet, there is a sense in which this transition from classical to quantum is incomplete. Traditionally, quantum Shannon theory has focused on scenarios where the internal states of information carriers are quantum, whilst their trajectories in spacetime have still been assumed to be classical. This work presents a second level of quantisation where both the information itself and its propagation in spacetime are treated in a quantum fashion. The second-quantised Shannon theory describes the possibility of a single particle being simultaneously transmitted through multiple communication channels in a quantum superposition. The framework is developed using the tools of higher-order transformations and quantum resource theories, formally quantifying the resources required for communication between a sender and receiver in this setting. The advantages of the second-quantised theory are illustrated in a series of examples, showcasing various counterintuitive phenomena that occur when information is simultaneously transmitted through multiple communication channels. In particular, when a single particle travels in a quantum superposition through two alternative transmission lines, the noisy processes in the two lines can destructively interfere, leading to a cleaner communication channel overall. Various different scenarios are encompassed in the framework, including transmission through a superposition of both independent and correlated channels, as well as through large-scale communication networks. This work concludes with a study of the robustness of these protocols to errors and a discussion of recent experimental demonstrations of their associated communication advantages.
A crucial step towards the 6th generation (6G) of networks would be a shift in communication paradigm beyond the limits of Shannon’s theory. In both classical and quantum Shannon’s information theory, communication channels are generally assumed to combine throughclassical trajectories, so that the associated network path traversed by the information carrier is well-defined. Counter-intuitively, quantum mechanics enables a quantum information carrier to propagate through aquantum path, i.e., through a path such that the causal order of the constituting communications channels becomes indefinite. Quantum paths exhibit astonishing features, such as providing non-null capacity even when no information can be sent through any classical path. In this paper, we study the quantum capacity achievable via a quantum path and establish upper and the lower bounds for it. Our findings reveal the substantial advantage achievable with a quantum path over any classical placements of communications channels in terms of ultimate achievable communication rates. Furthermore, we identify the region where a quantum path incontrovertibly outperforms the amount of transmissible information beyond the limits of conventional quantum Shannon’s theory, and we quantify this advantage over classical paths through a conservative estimate.
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.
This paper introduces a novel definition of non-classical information entropy, rooted in the principles of statistical physics, particularly quantum mechanics. The core premise is that traditional information entropy, derived from classical probability theory, is insufficient to capture the essence of information within non-classical systems exhibiting phenomena such as superposition and entanglement. We propose a definition based on the concept of accessible states and their associated probabilities, extending the notion of entropy beyond the realm of purely deterministic systems. The paper details the mathematical formulation of this non-classical information entropy, denoted as *Hnc*, and presents a method for its calculation. The method relies on identifying the set of accessible states for a given system and then applying a generalized form of Shannon's entropy formula, modified to account for the inherent uncertainties associated with quantum mechanics. The resulting *Hnc* provides a quantitative measure of the information content within non-classical systems, offering a potential framework for studying information processing and dynamics in quantum environments. The key contribution lies in bridging the gap between information theory and non-classical statistical physics, offering a new lens through which to understand information in quantum systems.