Abstract Here it is shown that quantum systems can be understood as information processors. Information and entropy are related quantities but also different, since the first is formal whilst the second is dynamical. Both quantum and classical information acquisition are a three-step process that needs a processor, a regulator, and a decider.
Abstract Here it is shown that quantum systems can be understood as information processors. Information and entropy are related quantities but also different, since the first is formal whilst the second is dynamical. Both quantum and classical information acquisition are a three-step process that needs a processor, a regulator, and a decider.
We show that the main difference between classical and quantum systems can be understood in terms of information entropy. Classical systems can be considered the ones where the internal dynamics can be known with arbitrary precision while quantum systems can be considered the ones where the internal dynamics cannot be accessed at all. As information entropy can be used to characterize how much the state of the whole system identifies the state of its parts, classical systems can have arbitrarily small information entropy while quantum systems cannot. This provides insights that allow us to understand the analogies and differences between the two theories.
Abstract This book deals with the basic concept of information, its importance, and the similarities and differences between classical and quantum information. First, classical information theory and its application to communication are discussed. Quantum mechanics and communication based on the laws of quantum mechanics are then considered. Quantum entanglement, a key resource in quantum communication, is analysed both from the fundamental perspective and from the perspective of its information-processing capability. This book also introduces quantum computation (in the context of the search and factorisation algorithms) and the basics of quantum error correction. Throughout this book, the connections between information theory, thermodynamics, and (quantum) physics are emphasised and discussed. Other topics covered in the book are data compression, related measures of information such as relative entropy, joint entropy, conditional entropy, and mutual information, along with the capacity of a noisy channel.
Quantum information processing is a modern research topic in which many different disciplines within and beyond physics interact. This chapter first makes contact with basic concepts of classical information theory, which is often not familiar to physicists. It then introduces Shannon's entropy and related entropic quantities, revisits the sampling theorem, discusses the relation between thermodynamics, measurement, and information theory, and looks into Landauer's principle. The chapter looks into the very basics of quantum information, the description of qubits, and operations of qubits. Finally, qubit implementations are discussed based on charge and spin qubits in quantum dots. Experimental results on single and two-qubit operations, and issues of decoherence are discussed.
License: CC BY-NC-ND 1. Introduction and epistemological framework Despite extraordinary empirical successes, modern natural science faces a fundamental explanatory gap. The General Theory of Relativity (GTR) describes the macroscopic structure of space, time and gravitation with unrivalled precision, whilst quantum mechanics (QM) governs the probabilistic laws of the microscopic world. However, both theoretical foundations are based on incompatible mathematical and ontological assumptions: GR assumes a continuous, deterministic space-time continuum; QM operates on discrete, non-local state spaces and is inextricably linked to the unsolved measurement problem. Holistic Information Theory (HIT) proposes a formal extension of the physical frame of reference. It begins one step prior to spatio-temporal localisation and poses the fundamental question regarding the a priori condition under which physical quantities can arise, be distinguished and measured in the first place. Every scientific description necessarily begins with a distinction; every distinction encodes a difference, and every difference is, at its core, pure information. From this methodological premise, GIT deduces that information is not a secondary product of biological evolution or technical data processing, nor is it merely a property of matter. Information is the primary ontological quantity of existence. Space, time, gravity, light, energy and matter are established as secondary forms of organisation and manifestation of an all-encompassing, evolutionary information process. In this system, cognition and physical manifestation are not passive processes, but active acts of information creation, initiated by a directed ‘question’ – understood as a physical interaction or systemic coupling.
We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.
We investigate fundamental connections between thermodynamics and quantum information theory. First, we show that the operational framework of thermal operations is nonequivalent to the framework of Gibbs-preserving maps, and we comment on this gap. We then introduce a fully information-theoretic framework generalizing the above by making further abstraction of physical quantities such as energy. It is technically convenient to work with and reproduces known results for finite-size quantum thermodynamics. With our framework we may determine the minimal work cost of implementing any logical process. In the case of information processing on memory registers with a degenerate Hamiltonian, the answer is given by the max-entropy, a measure of information known from quantum information theory. In the general case, we obtain a new information measure, the "coherent relative entropy", which generalizes both the conditional entropy and the relative entropy. It satisfies a collection of properties which justifies its interpretation as an entropy measure and which connects it to known quantities. We then present how, from our framework, macroscopic thermodynamics emerges by typicality, after singling out an appropriate class of thermodynamic states possessing some suitable reversibility property. A natural thermodynamic potential emerges, dictating possible state transformations, and whose differential describes the physics of the system. The textbook thermodynamics of a gas is recovered as well as the form of the second law relating thermodynamic entropy and heat exchange. Finally, noting that quantum states are relative to the observer, we see that the procedure above gives rise to a natural form of coarse-graining in quantum mechanics: Each observer can consistently apply the formalism of quantum information according to their own fundamental unit of information.