This is the first of two physically-oriented chapters in which we take a break from R-matrices and such topics and explore instead a completely different setting in which Hopf algebras arise naturally. This is the setting of logic, quantum mechanics and probability theory, in all of which contexts the coproduct Δ of the Hopf algebra plays the role of ‘sharing out’ possibilities. The theory has potential applications in computer science as well. In Boolean algebra and quantum mechanics, the product in the algebra corresponds in some sense to deduction of facts (‘putting two and two together’), and supplementing this structure with a coproduct provides a kind of reverse of this process by sharing out possible explanations of a fact. This provides the general theme of the present chapter. The chapter can also be viewed as background material for the next chapter, where we present a number of concrete models of quantum-mechanical Hopf algebras of observables. The simplest example of a combinatorial Hopf algebra is the shuffle algebra associated to an alphabet of symbols. Here the coproduct of a word is the formal sum of all pairs of words which could be shuffled to give the original word . Here, a shuffle of two words takes symbols from each of the words and interleaves them randomly while preserving the order of each word (this is exactly what you do when shuffling a pack of cards).
This is the first of two physically-oriented chapters in which we take a break from R-matrices and such topics and explore instead a completely different setting in which Hopf algebras arise naturally. This is the setting of logic, quantum mechanics and probability theory, in all of which contexts the coproduct Δ of the Hopf algebra plays the role of ‘sharing out’ possibilities. The theory has potential applications in computer science as well. In Boolean algebra and quantum mechanics, the product in the algebra corresponds in some sense to deduction of facts (‘putting two and two together’), and supplementing this structure with a coproduct provides a kind of reverse of this process by sharing out possible explanations of a fact. This provides the general theme of the present chapter. The chapter can also be viewed as background material for the next chapter, where we present a number of concrete models of quantum-mechanical Hopf algebras of observables. The simplest example of a combinatorial Hopf algebra is the shuffle algebra associated to an alphabet of symbols. Here the coproduct of a word is the formal sum of all pairs of words which could be shuffled to give the original word . Here, a shuffle of two words takes symbols from each of the words and interleaves them randomly while preserving the order of each word (this is exactly what you do when shuffling a pack of cards).
Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated with a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realized in Lin(H) in such a way that Δh=W(h⊗1)W−1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantum-mechanically to the system at time t+δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a novel kind of CTP theorem.
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.
Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable. Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.
Quantum computation and quantum computational logics give rise to some non-standard probability spaces that are interesting from a formal point of view. In this framework, events represent quantum pieces of information (qubits, quregisters, mixtures of quregisters), while operations on events are identified with quantum logic gates (which correspond to dynamic reversible quantum processes). We investigate the notion of Shi–Aharonov quantum computational algebra. This structure plays the role for quantum computation that is played by σ-complete Boolean algebras in classical probability theory.
Abstract Monoidal category theory serves as a powerful framework for describing logical aspects of quantum theory, giving an abstract language for parallel and sequential composition and a conceptual way to understand many high-level quantum phenomena. Here, we lay the foundations for this categorical quantum mechanics, with an emphasis on the graphical calculus that makes computation intuitive. We describe superposition and entanglement using biproducts and dual objects, and show how quantum teleportation can be studied abstractly using these structures. We investigate monoids, Frobenius structures and Hopf algebras, showing how they can be used to model classical information and complementary observables. We describe the CP construction, a categorical tool to describe probabilistic quantum systems. The last chapter introduces higher categories, surface diagrams and 2-Hilbert spaces, and shows how the language of duality in monoidal 2-categories can be used to reason about quantum protocols, including quantum teleportation and dense coding. Previous knowledge of linear algebra, quantum information or category theory would give an ideal background for studying this text, but it is not assumed, with essential background material given in a self-contained introductory chapter. Throughout the text, we point out links with many other areas, such as representation theory, topology, quantum algebra, knot theory and probability theory, and present nonstandard models including sets and relations. All results are stated rigorously and full proofs are given as far as possible, making this book an invaluable reference for modern techniques in quantum logic, with much of the material not available in any other textbook.