Entanglement and Information in Algebraic Quantum Theories

Abstract

The algebraic approach to physical theories provides a general framework encompassing both classical and quantum mechanics. Accordingly, by looking at the behaviour of the relevant algebras of observables one can investigate structural and conceptual differences between the theories. Interesting foundational questions can be formulated algebraically and their answers are then given in a mathematically compelling way. My dissertation focuses on some philosophical issues concerning entanglement and quantum information as they arise in the algebraic approach. The se two concepts are connected in that one can exploit the non-local character of quantum theory to construct protocols of information theory which are not realized in the classical world. I first introduce the basic concepts of the algebraic formalism by reviewing von Neumann's work on the mathematical foundations of quantum theories. After presenting the reasons why von Neumann abandoned the standard Hilbert space formulation in favour of the algebraic approach, I show how his axiomatic program remained a mathematical "utopia" in mathematical physics. The Bayesian interpretation of quantum mechanics is grounded in information-theoretical considerations. I take on some specific problems concerning the extension of Bayesian statistical inference in infinite dimensional Hilbert space. I demonstrate that the failure of a stability condition, formulated as a rationality constraint for classical Bayesian conditional probabilities, does not undermine the Bayesian interpretation of quantum probabilities. I then provide a solution to the problem of Bayesian noncommutative statistical inference in general probability spaces. Furthermore, I propose a derivation of the a priori probability state in quantum mechanics from symmetry considerations. Finally, Algebraic Quantum Field Theory offers a rigorous axiomatization of quantum field theory, namely the synthesis of quantum mechanics and special relativity. In such a framework one can raise the question of whether or not quantum correlations are made stronger by adding relativistic constraints. I argue that entanglement is more robust in the relativistic context than in ordinary quantum theory. In particular, I show how to generalize the claim that entanglement across space-like separated regions of Minkowski spacetime would persist, no matter how one acts locally.