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## Handbook of Quantum Logic and Quantum Structures | ScienceDirect

We then discuss a characterization of quantum measures in terms of signed product measures. We next define a quantum integral and discuss some of its properties. Quantum Logic on finite dimensional Hilbert spaces slides The subspaces of a finite dimensional Hilbert space form a modular ortholattice, and have additional nice properties. The extra structure has consequences in the Birkhoff-Von Neumann propositional quantum logic of a finite dimensional Hilbert space.

For instance, the logic is decidable which is not true with a general modular lattice , and the set of tautologies one obtains depends on the dimension of the underlying Hilbert space.

I'll discuss these and related phenomena and possible connections to quantum computing. Orthomodularity and decompositions Orthomodularity arises from consideration of the closed subspaces of a Hilbert space, and it was long thought that orthomodularity was closely tied to the Hilbert space structure.

This is not the case.

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The closed subspaces of a Hilbert space correspond to direct product decompositions of the space, and the direct product decompositions of most familiar objects form orthomodular structures. This holds in particular for sets, groups, topological spaces, and so forth. The connection between orthomodularity and direct product decompositions is used to build an axiomatic system for experimental systems. Further, by their nature, direct product decompositions are creatures of an essentially categorical nature.

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This allows one to construct orthomodular structures from decompositions of abstract objects in quite general categories. In particular, categories currently considered in categorical approaches to quantum computation allow this approach. This yields a bridge between these newer categorical approaches and the quantum logic.

Proof Nets and Formal Feynman Diagrams I will describe a formal connection between proof nets in linear logic and Feynman diagrams.

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We introduce a calculus, the phi-calculus, which is inspired by Feynman diagrams in quantum field theory. The ingredients are formal integrals, formal power series, a derivative construct and analogues of the Dirac delta function. Many of the manipulations of proof nets can be understood as manipulations of formulas reminiscent of a beginning calculus course.

In particular the "box" behaves like an exponential. All the equations for MELL multiplicative-exponetial linear logic hold but I will not prove that in the talk. I will end with some speculations about Feynman diagrams. Expressing Quantum Computations in the Arrow Calculus We show that the model of quantum computation based on density matrices and superoperators can be decomposed in a pure classical functional part and an effectful part modeling probabilities and measurement.

The effectful part can be modeled using a generalization of monads called arrows. We express quantum computations with measurements using the arrow calculus extended with monadic constructions. This framework expresses quantum programming using well-understood and familiar classical patterns for programming in the presence of computational effects. In addition, the five laws of the arrow calculus provide a convenient framework for equational reasoning about quantum computations that include measurements.

Dynamic and Epistemic Perspectives on Quantum Behavior My talk consists of two parts, representing the recent and on-going joint work with A. Baltag in the papers below. First I will concentrate on an improvement of the older results due to Piron, Soler, Mayet and others on the Hilbert-complete axiomatizations of algebraic quantum logic and present a dynamic-logical setting, in which physical actions and not only static physical properties are logically represented. Secondly I will focus on compound systems and analyse both classical and quantum correlations. We use this second setting to investigate the relationship between the information carried by each of the parts of a complex system and the information carried by the whole system.

Zhenghan Wang Microsoft Station Q: Symmetry and composition in probabilistic theories Both classical and quantum mechanics offer a standard device wherewith to combine physical systems: If we are aiming to understand QM, we surely want to account for the existence of a single, canonical product affording both kinds of entanglement. In this talk, I describe a recipe for building probabilistic theories having strong symmetry properties, using as data any uniform method for extending the symmetry group of a classical system here: Subject to some plausible conditions, this construction leads to a symmetric monoidal category, in which the product is non-signaling.

There is no guarantee that the composite systems obtained in this way will support arbitrary product states; however, when this desideratum is met, good things follow. For the latter structures the term quantum structures is appropriate. The chapters of this Handbook, which are authored by the most eminent scholars in the field, constitute a comprehensive presentation of the main schools, approaches and results in the field of quantum logic and quantum structures.

Much of the material presented is of recent origin representing the frontier of the subject. The present volume focuses on quantum structures.

## Handbook of Quantum Logic and Quantum Structures: Quantum Structures

Among the structures studied extensively in this volume are, just to name a few, Hilbert lattices, D-posets, effect algebras MV algebras, partially ordered Abelian groups and those structures underlying quantum probability. We use cookies to help provide and enhance our service and tailor content and ads. By continuing you agree to the use of cookies. Kurt Engesser, Dov M.