Operational Quantum Theory I

Leevi Leppäjärvi

Convex operational theories form a class of physical theories that are built on the operational mixing of states of the system resulting in convex state spaces. Following the operational approach to describe physical experiments, the other operational concepts, such as measurements and state transformations, rise from the properties of the state space. In addition to quantum theory, the convex operational theories include but are not restricted to classical theories and quantum theory of processes. In the light of recent deep interest in quantum information theory, convex operational theories serve as means to consider information-theoretic principles in a more abstract framework. This allows to compare different types of theories against each other and further study the nature of these principles. Some of these principles can then even be used for different axiomatizations of quantum theory. This thesis serves to introduce the mathematical concepts related to convex operational theories and then use them to construct this class of theories in the ordered vector space formalism. We use the constructed class of theories to consider the most important aspects of the theories with applications in physical theories such as quantum theory. We study some of the most important non-classical properties of quantum theory in the more abstract framework of convex operational theories including original research on one of these features.

Heinrich Saller

Leevi Leppäjärvi

Convex operational theories form a class of physical theories that are built on the operational mixing of states of the system resulting in convex state spaces. Following the operational approach to describe physical experiments, the other operational concepts, such as measurements and state transformations, rise from the properties of the state space. In addition to quantum theory, the convex operational theories include but are not restricted to classical theories and quantum theory of processes. In the light of recent deep interest in quantum information theory, convex operational theories serve as means to consider information-theoretic principles in a more abstract framework. This allows to compare different types of theories against each other and further study the nature of these principles. Some of these principles can then even be used for different axiomatizations of quantum theory. This thesis serves to introduce the mathematical concepts related to convex operational theories and then use them to construct this class of theories in the ordered vector space formalism. We use the constructed class of theories to consider the most important aspects of the theories with applications in physical theories such as quantum theory. We study some of the most important non-classical properties of quantum theory in the more abstract framework of convex operational theories including original research on one of these features.

J. Brooke

Paul Busch, Marian Grabowski, PEKKA J. LAHTI

Uwe Franz, Marius Junge, Gilles Pisier +1 more

Operator space theory was the joining element of our workshop. This theory is a quantisation of the theory of Banach spaces. The talks at our meeting investigated the interactions of operator space theory with operator algebras and other areas of functional analysis, with classical and quantum probability, with noncommutative harmonic analysis, and, in particular, with quantum information theory.