Abstract Complementarity is a property of a pair of observables being ‘maximally distinct’ from each other and, in this chapter, we analyse this property in categorical terms as a pair of interacting Frobenius structures. Complementary observables play a central role in quantum information theory, and we will see how they can be used to understand the structure of the Deutsch—Jozsa algorithm. We show that complementarity is closely linked to the theory of Hopf algebras. We discuss how many-qubit gates can be modelled using only complementary Frobenius structures, such as controlled negation, controlled phase gates and arbitrary single qubit gates. This leads to the ZX calculus, a sound and complete way to handle quantum computations using only equations in the graphical calculus.
Abstract Complementarity is a property of a pair of observables being ‘maximally distinct’ from each other and, in this chapter, we analyse this property in categorical terms as a pair of interacting Frobenius structures. Complementary observables play a central role in quantum information theory, and we will see how they can be used to understand the structure of the Deutsch—Jozsa algorithm. We show that complementarity is closely linked to the theory of Hopf algebras. We discuss how many-qubit gates can be modelled using only complementary Frobenius structures, such as controlled negation, controlled phase gates and arbitrary single qubit gates. This leads to the ZX calculus, a sound and complete way to handle quantum computations using only equations in the graphical calculus.
This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX -calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z - and X -spin observables, which yields a scaled variant of a bialgebra.
Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.
Abstract A Frobenius structure is a monoid together with a comonoid, which satisfies an interaction law. Frobenius structures have a powerful graphical calculus and we prove a normal form theorem that makes them easy to work with. The Frobenius law itself is justified as a coherence property between daggers and closure of a category. We prove classification theorems for dagger Frobenius structures: in Hilb in terms of operator algebras and in Rel in terms of groupoids. Of special interest is the commutative case—as for Hilbert spaces this corresponds to a choice of basis—and provides a powerful tool to model classical information. We discuss phase gates and the state transfer protocol—as well as modules for Frobenius structures—and show how we can use these to model measurement, controlled operations and quantum teleportation.
Observables in quantum mechanics are generally complementary, that is, they reveal mutually incompatible pieces of information about a given system. This property is not only a fundamental tenet of the quantum formalism, but also a key component in quantum cryptographic protocols. As such it has fuelled much research into finding sets of highly complementary observables. In its strongest form -- the one we consider in this work -- the information between complementary observables is not merely incompatible but mutually exclusive: maximal information about one observable implies no information about the other, and vice versa. Maximal sets of non-degenerate complementary observables are known to exist in systems of prime power dimension such as n-qubit systems. Here, we study complementarity of degenerate observables, specifically we prove that observables associated with the n-qubit Pauli group also exhibit complementarity under this restriction: first, we obtain a criterion for two such observables to be complementary and, second, we relate maximal sets of complementary observables with informational pure state complementarity equalities, further studied in two companion papers. Equivalently, these results can be formulated in terms of maximal complementary sets of (not necessarily maximal) Abelian subgroups of the Pauli group linking with (possibly coarse-grained) mutually unbiased bases. Finally, we prove that these complementarity sets entail strong entropic uncertainty relations.
Categorical quantum mechanics places finite-dimensional quantum theory in the context of compact closed categories, with an emphasis on diagrammatic reasoning. In this framework, two equational diagrammatic calculi have been proposed for pure-state qubit quantum computing: the ZW calculus, developed by Coecke, Kissinger and the first author for the purpose of qubit entanglement classification, and the ZX calculus, introduced by Coecke and Duncan to give an abstract description of complementary observables. Neither calculus, however, provided a complete axiomatisation of their model.
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.