Quantum Foundations

A very general way of describing the abstract structure of quantum theory is to say that the set of observables on a quantum system form a C*-algebra.  A natural question is then, why should this be the case - why can observables be added and multiplied together to form any algebra, let alone of the special C* variety?  I will present recent work with Markus Mueller and Howard Barnum, showing that the closest algebraic cousins to standard quantum theory, namely the Jordan-algebras, can be characterized by three principles having an informational flavour, namely: (1) a generalized spectral decomposition, (2) a high degree of symmetry, and (3) a requirement on conditioning on the results of observations.   I'll then discuss alternatives to the third principle, as well as the possibility of dropping it as a way of searching for natural post-quantum theories.

It is certainly possible to express ordinary quantum mechanics in the framework of a real vector space: by adopting a suitable restriction on all operators--Stueckelberg’s rule--one can make the real-vector-space theory exactly equivalent to the standard complex theory.  But can we achieve a similar effect without invoking such a restriction?  In this talk I explore a model within real-vector-space quantum theory in which the role of the complex phase is played by a separate physical system called the ubit (for “universal rebit”).  The ubit is a single binary real-vector-space quantum object that is allowed to interact with everything in the world.  It also rotates in its two-dimensional state space.  In the limit of infinitely fast rotation, one recovers standard quantum theory.  When the rotation rate is large but not infinite, one finds small deviations from the standard theory.  Here I describe a few such deviations that we have seen numerically and explained analytically.

The distinction between a realist interpretation of quantum theory that is psi-ontic and one that is psi-epistemic is whether or not a difference in the quantum state necessarily implies a difference in the underlying ontic state. Psi-ontologists believe that it does, psi-epistemicists that it does not. This talk will address the question of whether the PBR theorem should be interpreted as lending evidence against the psi-epistemic research program. I will review the evidence in favour of the psi-epistemic approach and describe the pre-existing reasons for thinking that if a quantum state represents knowledge about reality then it is not reality as we know it, i.e., it is not the kind of reality that is posited in the standard hidden variable framework. I will argue that the PBR theorem provides additional clues for "what has to give" in the hidden variable framework rather than providing a reason to retreat from the psi-epistemic position. The first assumption of the theorem - that holistic properties may exist for composite systems, but do not arise for unentangled quantum states - is only appealing if one is already predisposed to a psi-ontic view. The more natural assumption of separability (no holistic properties) coupled with the other assumptions of the theorem rules out both psi-ontic and psi-epistemic models and so does not decide between them. The connection between the PBR theorem and other no-go results will be discussed. In particular, I will point out how the second assumption of the theorem is an instance of preparation noncontextuality, a property that is known not to be achievable in any ontological model of quantum theory, regardless of the status of separability (though not in the form posited by PBR). I will also consider the connection of PBR to the failure of local causality by considering an experimental scenario which is in a sense a time-inversion of the PBR scenario.