In this chapter we provide an overview of the new principles used for reconstructing quantum theory. We have already seen the principles in Chapter 2, where they have been derived as theorems of the quantum theory. In Part IV of this book we will actually do the reverse process, namely to derive quantum theory from the principles. All features of quantum theory – ranging from the superposition principle, entanglement, no cloning, teleportation, Bell's inequalities violation, quantum cryptography – can be understood and proved using only the principles, without using Hilbert spaces, and this is indeed the aim of Part III of the book, whereas Part IV contains the derivation of the theory from the principles. All the six principles are operational, in that they stipulate whether or not certain tasks can be accomplished: they set the rules of the game for all the experiments and all the protocols that can be carried out in the theory. They also provide a great insight into the worldview at which quantum theory hints. We review the list of the principles: 1. Atomicity of composition 2. Perfect discriminability 3. Ideal compression 4. Causality 5. Local discriminability 6. Purification All six principles, with the exception of purification, express standard features that are shared by both classical and quantum theory. The principle of purification picks up uniquely quantum theory among the theories allowed by the first five, partly explaining the magic of quantum information. The principles of causality, local discriminability, and purification will be thoroughly analyzed in Chapters 5, 6, and 7, respectively. The other three principles, atomicity of composition, perfect discriminability, ideal compression, are examined in the remainder of the present chapter. Atomicity of Composition In the general framework we encountered the notions of coarse-grained and atomic operation. A coarse-grained operation is obtained by joining together outcomes of a test, corresponding to neglect some information. The inverse process of coarse-graining is that of refining. An atomic operation is one where no information has been neglected, namely an operation that cannot be refined. When the operation is atomic, the experimenter has maximal knowledge of what's happening in the lab. A test consisting of atomic operations represents the highest level of control achievable according to our theory. But, is it possible to maintain such a level of control throughout a sequence of experiments?
In this chapter we provide an overview of the new principles used for reconstructing quantum theory. We have already seen the principles in Chapter 2, where they have been derived as theorems of the quantum theory. In Part IV of this book we will actually do the reverse process, namely to derive quantum theory from the principles. All features of quantum theory – ranging from the superposition principle, entanglement, no cloning, teleportation, Bell's inequalities violation, quantum cryptography – can be understood and proved using only the principles, without using Hilbert spaces, and this is indeed the aim of Part III of the book, whereas Part IV contains the derivation of the theory from the principles. All the six principles are operational, in that they stipulate whether or not certain tasks can be accomplished: they set the rules of the game for all the experiments and all the protocols that can be carried out in the theory. They also provide a great insight into the worldview at which quantum theory hints. We review the list of the principles: 1. Atomicity of composition 2. Perfect discriminability 3. Ideal compression 4. Causality 5. Local discriminability 6. Purification All six principles, with the exception of purification, express standard features that are shared by both classical and quantum theory. The principle of purification picks up uniquely quantum theory among the theories allowed by the first five, partly explaining the magic of quantum information. The principles of causality, local discriminability, and purification will be thoroughly analyzed in Chapters 5, 6, and 7, respectively. The other three principles, atomicity of composition, perfect discriminability, ideal compression, are examined in the remainder of the present chapter. Atomicity of Composition In the general framework we encountered the notions of coarse-grained and atomic operation. A coarse-grained operation is obtained by joining together outcomes of a test, corresponding to neglect some information. The inverse process of coarse-graining is that of refining. An atomic operation is one where no information has been neglected, namely an operation that cannot be refined. When the operation is atomic, the experimenter has maximal knowledge of what's happening in the lab. A test consisting of atomic operations represents the highest level of control achievable according to our theory. But, is it possible to maintain such a level of control throughout a sequence of experiments?
It is not surprising that our language should be incapable of describing processes occurring within atoms, for, as has been remarked, it was invented to describe the experiences of daily life, … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme–the quantum theory–which seems entirely adequate for the treatment of atomic processes. Werner Heisenberg, The Physical Principles of the Quantum Theory (1930) In this chapter we introduce the mathematical structure of quantum theory, for a finite number of systems and finite dimension, starting from three simple Hilbert-space postulates, and based on the intuitive notions of state, transformation, and effect, within a causal context. Such notions will be formalized in a general non-causal context in Chapter 3, which is devoted to the informational framework. We will assume the reader has a basic knowledge of matrix analysis and of operators over Hilbert spaces. Comments about the extension to infinite dimensions, continuous outcome-spaces, and infinitely many systems can be found in the notes at the end of the chapter. From the three simple Hilbert-space postulates, the present chapter extracts six principles, which will become the postulates for quantum theory in the axiomatic derivation presented in Part III of the book. In addition, we will take the opportunity to present and explore many relevant features of the theory – including its convex and causal structure, the Choi–Jamiołkowski isomorphism, and relevant notions, such as those of POVM, complete positivity of maps, quantum operation and channels, entangled states, state discrimination, tomography of states, and transformations. The purification principle will open the stage to the dilation theorems of the theory of quantum open systems, along with other consequences, such as the existence of faithful and steering states, the theorem of no information without disturbance, quantum teleportation, and more. We will review the quantum no-cloning and no-programming theorems, and finally provide a thorough derivation of the theory of quantum channel inversion, which is the backbone of the general theory of quantum error correction in quantum information. Additional interesting topics will be classical information theory as a restriction of quantum information theory, and a critical reconsideration of the von Neumann postulate and the notion of observable.
What makes quantum theory so special? Everyone has an opinion on that: one can point at entanglement, non-locality, complementarity, uncertainty, and many other features. In a trivial sense, every answer is legitimate: every feature in which quantum theory differs from other familiar theories can be taken as the distinctive feature, because it allows you to tell quantum theory apart from what you already know. But this would not take you very far. The real question is not how to distinguish quantum theory from theories that you already know, but how to distinguish it from all theories that you could possibly imagine. Most features are not good at that – for example, physicists have invented many toy theories that exhibit entanglement, non-locality, and complementarity and still are not quantum theory. In this chapter we put forward our answer to the question of what makes quantum theory special: purification. In the following, we will explore some of the most fundamental facts about quantum theory, like entanglement and teleportation, connecting purification with a deep stream of thought that goes back to Schrödinger and von Neumann. The results presented in this chapter are also a first example of how one can reason on quantum theory starting from first principles. A Distinctive and Fundamental Trait The purification principle stipulates that, whenever you are ignorant about the state of a system A, you can always claim that your ignorance comes from the fact that A is a part of a larger system AB, of which you have full knowledge. When you do this, the pure state that you have to assign to the composite system AB is determined by the state of A in an essentially unique way. The purification of mixed states is a peculiar feature – surely, not one that we experience in our everyday life. How can you claim that you know A and B if you do not know A alone? This counterintuitive feature had been noted already in the early days of quantum theory, when Erwin Schrödinger famously wrote: “Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts .” And, in the same paper: “I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”
Quantum theory is the soul of theoretical physics. It is not just a theory of specific physical systems, but rather a new framework with universal applicability. This book shows how we can reconstruct the theory from six information-theoretical principles, by rebuilding the quantum rules from the bottom up. Step by step, the reader will learn how to master the counterintuitive aspects of the quantum world, and how to efficiently reconstruct quantum information protocols from first principles. Using intuitive graphical notation to represent equations, and with shorter and more efficient derivations, the theory can be understood and assimilated with exceptional ease. Offering a radically new perspective on the field, the book contains an efficient course of quantum theory and quantum information for undergraduates. The book is aimed at researchers, professionals, and students in physics, computer science and philosophy, as well as the curious outsider seeking a deeper understanding of the theory.
Quantum theory is the soul of theoretical physics. It is not just a theory of specific physical systems, but rather a new framework with universal applicability. This book shows how we can reconstruct the theory from six information-theoretical principles, by rebuilding the quantum rules from the bottom up. Step by step, the reader will learn how to master the counterintuitive aspects of the quantum world, and how to efficiently reconstruct quantum information protocols from first principles. Using intuitive graphical notation to represent equations, and with shorter and more efficient derivations, the theory can be understood and assimilated with exceptional ease. Offering a radically new perspective on the field, the book contains an efficient course of quantum theory and quantum information for undergraduates. The book is aimed at researchers, professionals, and students in physics, computer science and philosophy, as well as the curious outsider seeking a deeper understanding of the theory.
We review a recent approach to the foundations of quantum mechanics inspired by quantum information theory [1, 2]. The approach is based on a general framework, which allows one to address a large class of physical theories which share basic information-theoretic features. We first illustrate two very primitive features, expressed by the axioms of causality and purity-preservation, which are satisfied by both classical and quantum theory. We then discuss the axiom of purification, which expresses a strong version of the Conservation of Information and captures the core of a vast number of protocols in quantum information. Purification is a highly non-classical feature and leads directly to the emergence of entanglement at the purely conceptual level, without any reference to the superposition principle. Supplemented by a few additional requirements, satisfied by classical and quantum theory, it provides a complete axiomatic characterization of quantum theory for finite dimensional systems.