Final Report Summary - QCLS (Quantum Computation, Logic, and Security)
The main `product' of the project is: effectus theory. It is a new branch of categorical logic that aims to capture the essentials of quantum logic, with probabilistic and Boolean logic as special cases. Predicates in effectus theory are not subobjects having a Heyting algebra structure, like in topos theory, but `characteristic' functions, forming effect algebras. Such effect algebras are algebraic models of quantitative logic, in which double negation holds. Effects in quantum theory and fuzzy predicates in probability theory form examples of effect algebras. The main quantum instantiation of the notion of effectus is given by von Neumann algebras. Four of the six PhD theses that come out of this ERC project contribute to effectus theory.
The theory of effectuses includes the fundamental duality between states and effects, with the associated Born rule for validity of an effect (predicate) in a particular state. A basic result says that
effectuses can be described equivalently in both `total' and `partial' form. So-called `commutative' and `Boolean' effectuses are distinguished, for probabilistic and classical models respectively. It
has been shown that these Boolean effectuses are essentially extensive categories. Part of the theory is devoted to the logical notions of comprehension and quotient, which are described abstractly as right adjoint to truth, and as left adjoint to falisity, respectively. Comprehension and quotients form the basis for the identification of pure maps and measurements in effectus theory, and also for a dagger operation that captures the intrinsice reversible charachter of pure quantum operations.
This theory of effectuses, and the underlying notions of effect algebra and effect module, plays an important role in other work done within the project, such as quantum type/domain theory and a both classical & quantum probabilistic Bayesian reasoning (including causality) and in a new approach to contextuality and non-locality.
The theory of effectuses includes the fundamental duality between states and effects, with the associated Born rule for validity of an effect (predicate) in a particular state. A basic result says that
effectuses can be described equivalently in both `total' and `partial' form. So-called `commutative' and `Boolean' effectuses are distinguished, for probabilistic and classical models respectively. It
has been shown that these Boolean effectuses are essentially extensive categories. Part of the theory is devoted to the logical notions of comprehension and quotient, which are described abstractly as right adjoint to truth, and as left adjoint to falisity, respectively. Comprehension and quotients form the basis for the identification of pure maps and measurements in effectus theory, and also for a dagger operation that captures the intrinsice reversible charachter of pure quantum operations.
This theory of effectuses, and the underlying notions of effect algebra and effect module, plays an important role in other work done within the project, such as quantum type/domain theory and a both classical & quantum probabilistic Bayesian reasoning (including causality) and in a new approach to contextuality and non-locality.