Foundations of Quantum Mechanics
The past decade or so has produced a handful of derivations, or reconstructions, of finite-dimensional quantum mechanics from various packages of operational and/or information-theoretic principles. I will present a selection of these principles --- including symmetry postulates, dilational assumptions, and versions of Hardy's subspace axiom --- in a common framework, and indicate several ways, some familiar and some new, in which these can be combined to yield either standard complex QM (with or without SSRs) or broader theories embracing formally real Jordan algebras.
The device-independent approach to physics is one where conclusions are drawn directly and solely from the observed correlations between measurement outcomes. This operational approach to physics arose as a byproduct of Bell's seminal work to distinguish quantum correlations from the set of correlations allowed by locally-causal theories. In practice, since one can only perform a finite number of experimental trials, deciding whether an empirical observation is compatible with some class of physical theories will have to be carried out via the task of hypothesis testing.
To identify which principles characterise quantum correlations, it is essential to understand in which sense this set of correlations differs from that of almost quantum correlations. We solve this problem by invoking the so-called no-restriction hypothesis, an explicit and natural axiom in many reconstructions of quantum theory stating that the set of possible measurements is the dual of the set of states.
In this status report on current work in progress, I will sketch a generalization of the temporal type theory introduced by Schultz and Spivak to a logic of space and spacetime. If one writes down a definition of probability space within this logic, one conjecturally obtains a notion whose semantics is precisely that of a Euclidean quantum field. I will sketch how to use the logic to reason about probabilities of events involving fields, sketch the relation to AQFT, and attempt to formulate the DLR equations within the logic.
I give further details on a unification of the foundations of operational quantum theory with those of quantum field theory, coming out of a program that is also known as the positive formalism. I will discuss status and challenges of this program, focusing on the central new concept of local quantum operation. Among the conceptual challenges I want to highlight the question of causality. How do we know that future choices of measurement settings do not influence present measurement results? Should we enforce this, as in the standard formulation of quantum theory?
Cellular automata are a central notion for the formulation of physical laws in an abstract information-theoretical scenario, and lead in recent years to the reconstruction of free relativistic quantum field theory. In this talk we extend the notion of a Quantum Cellular Automaton to general Operational Probabilistic Theories. For this purpose, we construct infinite composite systems, illustrating the main features of their states, effects and transformations.
From a brief discussion of how to generalise Reichenbach’s Principle of the Common Cause to the case of quantum systems, I will develop a formalism to describe any set of quantum systems that have specified causal relationships between them. This formalism is the nearest quantum analogue to the classical causal models of Judea Pearl and others. At the heart of the classical formalism lies the idea that facts about causal structure enforce constraints on probability distributions in the form of conditional independences.