# Quantum Theory from First Principles: An Informational Approach

Book Preface

The book is the result of 20 years of teaching and research by the three authors in the fields of quantum foundations and quantum information, which culminated in two long joint papers [Phys. Rev. A 81 062348 (2010) and 84 012311 (2011)] that derive quantum theory from six simple information-theoretical principles. We have now the opportunity of presenting quantum theory in a radically new way, based on a conceptual understanding from the new principles. By “quantum theory” we mean the general theory of physical systems that lies at the core of “quantum mechanics,” the latter broadly viewed as the quantum generalization of classical Hamiltonian mechanics. The book will not cover applications to “mechanics,” but rather focus on applications to quantum information.

For this reason, and with the aim of keeping the center of attention more on conceptual issues, rather than on the mathematical technicalities, we consider finite numbers of finite-dimensional systems, and restrict to probabilities of finite set of events, with some extensions to the infinite/continuous case discussed in the notes at the end of chapters.

The book includes 220 exercises and problems. The exercises are given within the body of each chapter, and selected solutions can be found at the end of that chapter. The exercises represent an integral part of the book and we warmly recommend the reader to solve them (or to check out the solutions), because the results proven therein are often used in our arguments. The problems presented at the end of each chapter build up additional knowledge and problem-solving skills, not strictly needed for the understanding of the arguments, but it is still recommended to solve them (or study the solutions). References, historical comments, and citations are provided in the notes at the end of chapters.

The book can be used for teaching at all levels, ranging from undergraduate, to master, up to PhD, and for pursuing personal research interests. The book is divided into four parts, organized as follows:

Part I The Status Quo (Chapter 2) introduces the mathematical structure of quantum theory, deriving it from three simple Hilbert-space postulates (systems, states, and the no-restriction hypothesis), and proving, in the form of theorems, what will later become our six principles for the derivation of quantum theory. The full mathematical structure of quantum theory is derived, including all relevant results in quantum open systems and quantum information. The derivation uses original powerful proving techniques based on tensor operators. In this part, the reader will have the chance to become acquainted with the relevant notions in operational probabilistic theories (OPT) and in convex analysis, and will start using the six principles for deriving results. This entire part can be used for an undergraduate semester course of quantum theory and quantum information, for physicists, mathematicians, and computer scientists.

Part II The Informational Approach (Chapters 3–7) presents the framework of operational probabilistic theories and introduces the six principles. Three separate chapters are devoted to the main principles of causality, local discriminability, and purification. Parts I and II together make a complete semester course for a masters-level course.

Part III Quantum Information Without Hilbert Spaces (Chapters 8–12) uses all the six principles to derive key results of quantum information theory and general features of quantum theory, including no-go theorems such as the no-cloning and no-bit-commitment theorems. Some parts of these chapters can be incorporated in a masters-level course.

Part IV Quantum Theory from the Principles (Chapters 13–17) derives quantum theory from the six principles.

Chapters on causality, local discriminability, and purification are of interest also for philosophers and, more generally, for readers who are seeking for a deeper understanding of these concepts in the light of quantum information.

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