We derive quantum theory from purely informational principles. Five elementary axioms---causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning---define a broad class of theories of information processing that can be regarded as standard. One postulate---purification---singles out quantum theory within this class.
We derive quantum theory from purely informational principles. Five elementary axioms---causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning---define a broad class of theories of information processing that can be regarded as standard. One postulate---purification---singles out quantum theory within this class.
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?
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.
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.
We propose four information-theoretic axioms for the foundations of statistical mechanics in general physical theories. The axioms---Causality, Purity Preservation, Pure Sharpness, and Purification---identify a class of theories where every mixed state can be modelled as the marginal of a pure entangled state and where every unsharp measurement can be modelled as a sharp measurement on a composite system. This class of theories, called sharp theories with purification, includes quantum theory both with complex and real amplitudes, as well as a suitable extension of classical probability theory where classical systems can be entangled with other, non-classical systems. Theories satisfying our axioms support well-behaved notions of majorization, entropy, and Gibbs states, allowing for an information-theoretic derivation of Landauer's principle. We conjecture that every theory admitting a sensible thermodynamics must be extendable to a sharp theory with purification.
There has been a body of work deriving the complex Hilbert-space structure of quantum theory from axioms/principles/postulates to deepen our understanding of quantum theory and to reveal ways to go beyond it to resolve foundational issues. Recent progress in incorporating indefinite causal structure into physical theories suggests that a more comprehensive understanding of both quantum theory and the theory beyond it accounts for indefinite causal structure. We formulate a framework of physical theories without assuming definite causal structure and identify postulates that single out the complex Hilbert-space structure. More than one complex Hilbert-space theory is compatible with the postulates, which leaves room for further search for the best among these theories.
After more than a century since its birth, Quantum Theory still eludes our understanding. If asked to describe it, we have to resort to abstract and ad hoc principles about complex Hilbert spaces. How is it possible that a fundamental physical theory cannot be described using the ordinary language of Physics? Here we offer a contribution to the problem from the angle of Quantum Information, providing a short non-technical presentation of a recent derivation of Quantum Theory from information-theoretic principles. The broad picture emerging from the principles is that Quantum Theory is the only standard theory of information that is compatible with the purity and reversibility of physical processes.