Conference

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. In this talk, I will review some recent progress on this task based on the prediction-based-ratio method and discuss how it may allow us to falsify, in principle, other classes of physical theories, such as those constrained only by the nonsignaling principle, and those that are constrained to produce the so-called "almost-quantum" set of correlations. As an application, I demonstrate how this method allows us to unveil the apparent violation of the nonsignaling conditions in certain experimental data collected in a Bell test. The lesson learned from this observation will be briefly discussed.

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. We prove that, contrary to quantum correlations, no generalised probabilistic theory satisfying the no-restriction hypothesis is able to reproduce the set of almost quantum correlations. Therefore, any theory whose correlations are exactly, or very close to, the almost quantum correlations necessarily requires a rule limiting the possible measurements. Our results suggest that the no-restriction hypothesis may play a fundamental role in singling out the set of quantum correlations among other non-signalling ones.

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. Joint work with David Spivak

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? Should this "emerge" from a fundamental theory? Does this question even make sense in a context without a fixed notion of time, such as quantum gravity? With a heavy dose of speculation (put also grounded in very concrete evidence) I find that fermionic theories might play an essential role.

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. I will describe a quantum analogue of this idea, which leads to a quantum version of the three rules of Pearl’s do-calculus. If time, I will end with some more speculative remarks concerning the significance of the work for the foundations of quantum theory.