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.”
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.”
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?
The astute reader might have formed the impression that quantum in formation science is a rather qualitative discipline because we have not, as yet, explained how to quantify quantum information. There are three good reasons for leaving this important question until the final chapter. Firstly, quantum information theory is technically demanding and to treat it at an earlier stage might have suggested that our subject was more complicated than it is. Secondly, there is the fact that many of the ideas in the field, such as teleportation and quantum circuits, are unfamiliar and it was important to present these as simply as possible. Finally, and most importantly, the theory of quantum information is not yet fully developed. It has not yet reached, in particular, the level of completeness of its classical counterpart. For this reason we can answer only some of the many questions we would like a quantum theory of information to address. Having said this, we can say that however, there are beautiful and useful mathematical results and it seems certain that these will continue to form an important part of the theory as it develops. We noted in the introduction to Chapter 1 that ‘quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed’. A presentation of at least the beginnings of a quantitative theory is the objective of this final chapter. The entropy or information derived from a given probability distribution is, as we have seen, a convenient measure of the uncertainty associated with the distribution. If many of the probabilities are large, so that many of the possible events are comparably likely, then the entropy will be large. If one probability is close to unity, however, then the entropy will be small. It is convenient to introduce entropy in quantum mechanics as a measure of the uncertainty, or lack of knowledge, of the form of the state vector. If we know that our system is in a particular pure state then the associated uncertainty or entropy should be zero. For mixed states, however, it will take a non-zero value.
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.
How it all began In 1935, a decade after the invention of quantum theory by Heisenberg (1925) and Schrödinger (1926), three papers appeared, setting the stage for a new concept, quantum entanglement. The previous decade had already seen momentous discussions on the meaning of the new theory, for example by Niels Bohr (see his contribution to Albert Einstein: Philosopher-Scientist in Further reading); but these new papers opened up the gates for a new, very deep philosophical debate about the nature of reality, about the role of knowledge, and about their relation. Since the 1970s, it has become possible for experimenters to observe entanglement directly in the laboratory and thus test the counter-intuitive predictions of quantum mechanics whose confirmation today is beyond reasonable doubt. Since the 1990s, the conceptual and experimental developments have led to new concepts in information technology, including such topics as quantum cryptography, quantum teleportation, and quantum computation, in which entanglement plays a key role. In the first of the three papers, Einstein, Podolsky, and Rosen (EPR) realized that, if two particles had interacted in their past, their properties will remain connected in the future in a novel way, namely, observation on one determines the quantum state of the other, no matter how far away. Their conclusion was that quantum mechanics is incomplete and they expressed their belief that a more complete theory may be found. Niels Bohr replied that the two systems may never be considered independent of each other, and observation on one changes the possible predictions on the other. Finally, Erwin Schrödinger, in the paper which also proposed his famous cat paradox, coined the term “entanglement” (in German Verschränkung ) for the new situation and he called it “not one, but the essential trait of the new theory, the one which forces a complete departure from all classical concepts.”
We have seen, in Section 2.5, how the superposition principle leads to the existence of entangled states of two or more quantum systems. Such states are characterized by the existence of correlations between the systems, the form of which cannot be satisfactorily accounted for by any classical theory. These have played a central role in the development of quantum theory since early in its development, starting with the famous paradox or dilemma of Einstein, Podolsky, and Rosen (EPR). No less disturbing than the EPR dilemma is the problem of Schrödinger’s cat, an example of the apparent absurdity of following entanglement into the macroscopic world. It was Schrödinger who gave us the name entanglement; he emphasized its fundamental significance when he wrote, ‘I would call this not one but the characteristic trait of quantum mechanics, the one that enforces the entire departure from classical thought’. The EPR dilemma represents a profound challenge to classical reasoning in that it seems to present a conflict between the ideas of the reality of physical properties and the locality imposed by the finite velocity of light. This challenge and the developments that followed have served to refine the concept of entanglement and will be described in the first section of this chapter. We start by recalling that a state of two quantum systems is entangled if its density operator cannot be written as a product of density operators for the two systems, or as a probability-weighted sum of such products. For pure states, the condition for entanglement can be stated more simply: a pure state of two quantum systems is not entangled only if the state vector can be written as a product of state vectors for the two systems. In the discipline of quantum information, entanglement is viewed as a resource to be exploited. We shall find, both here and in the subsequent chapters, that our subject owes much of its distinctive flavour to the utilization of entanglement.
In classical mechanics, two (or more) systems that do not interact – that do not exert any kind of force on each other – are completely separated (see Sec. 1.1), i.e. experiments performed locally on one of them cannot in any way influence experiments performed locally on the other. Quantum mechanics, on the other hand, admits that there can be a form of interdependence even in absence of physical interaction. This type of correlation goes under the name of entanglement (see Def. 5.1: p. 183). We already know that entanglement means non-factorizability of the state vector describing a system with at least two different degrees of freedom. In this chapter, we shall see that this may be the case also for systems that are very far away (space-like distant) from each other and we shall learn many interesting consequences of this state of affairs. Recent developments show that one may entangle even particles that have never interacted with each other. The starting point of the following analysis is necessarily represented by the pioneering paper of Einstein, Podolsky, and Rosen, who inaugurated an incredible series of theoretical and experimental investigations right up to the present day that have confirmed the predictions of quantum mechanics. The “irony” of this history is that Einstein, Podolsky, and Rosen, EPR for short, aimed to present a definitive proof of the inconsistency of quantum mechanics relative to classical physical theory. In fact, they initiated one of the most prolific fields of modern science.