Quantum Foundations

The simplest algebraic curves of genus one are the nonsingular cubics in two-dimensional complex projective space. Interpreting CP^2 as the space of pure quantum states associated with a Hilbert space of dimension three, I will show how various properties of d=3 symmetric informationally complete positive operator valued measures can be understood in terms of the geometry of such curves. The resulting structure, although of considerable complexity, is very beautiful from a mathematical perspective.

The solution of many problems in quantum information is critically dependent on the geometry of the space of density matrices. For a Hilbert space of dimension 2 this geometry is very simple: it is simply a sphere. However for Hilbert spaces of dimension greater than 2 the geometry is much more interesting as the bounding hypersurface is both highly symmetric (it has a d^2 real parameter symmetry group, where d is the dimension) and highly convoluted. The problem of getting a better understanding of this hypersurface is difficult (it is hard even in the case of a single qutrit). It is also, we believe, both physically important and mathematically deep. In this talk we relate the problem to MUBs (mutually unbiased bases) and SIC-POVMs (symmetric informationally complete positive operator valued measures). These structures were originally introduced in connection with tomography. However, that by no means exhausts their importance. In particular their existence (non-existence???) in a given dimension is a source of significant insight into the state space geometry in that dimension. SIC-POVMs are especially important in this regard as they provide a a natural set of coordinates for state space. In this talk we give an overview of the problem. We then go on to describe some recent results obtained in collaboration with Chris Fuchs and Hoan Dang (also see recent work by Wootters and Sussman). In particular we describe the connection with minimum uncertainty states. These states, besides being interesting in themselves (they are a kind of discrete analogue of coherent states with important cryptographic applications), suggest a potentially fruitful line of attack on the still outstanding SIC existence problem.

The mathematical formalism of quantum theory has many features whose physical origin remains obscure. In this paper, we attempt to systematically investigate the possibility that the concept of information may play a key role in understanding some of these features. We formulate a set of assumptions, based on generalizations of experimental facts that are representative of quantum phenomena and physically comprehensible theoretical ideas and principles, and show that it is possible to deduce the finite-dimensional quantum formalism from these assumptions. The concept of information, via an information-theoretic invariance principle, plays a central role in the derivation, and gives rise to some of the central structural features of the quantum formalism.

Collapse models are one of the most promising attempts to overcome the measurement problem of quanum mechanics: they descibe, within one single framework, both the quantum properties of microscopic systems and the classical properties of macroscopic objects, and in particular they explain why measurements always have definite outcomes, distributed according to the Born probability rule. We will discuss some recent developments in this field: i) we will show how it is possible to formulate collapse models in such a way that the mean energy of physical system does non increse indefinitely, a typical feature of the models first proposed in the literature; ii) we will discuss recent experiments aiming at testing the validity of the superposition principle, thus of collapse models, at the mesoscopic level.

We will postulate a novel notion of probability; this will involve introducing an extra axiom of probability that seems natural from a Bayesian perspective. We will then provide an analogue of Gleason's theorem for these probabilities. We will also discuss why this approach may be useful for generalizations of quantum theory such as quantum gravity theories; this will involve discussing an analogy between Bayesian approaches and relational approaches.

The problem of associating beables (hidden variables) to QFT, in the spirit of what Bohm did for nonrelativistic QM, is not trivial. In 1984, John Bell suggested a way of solving the problem, according to which the beables are the positions of fermions, in a discretized version of QFT, and obey a stochastic evolution that simulates all predictions of QFT. In the continuum limit, it will be shown that the Bell model becomes deterministic and that it is related to the choice of the charge density as a beable. Moreover, the charge superselection rule is a consequence of the Bell model. The non-relativistic limit and the derivation of Bohm's first quantized interpretation in this limit are also studied. I will also consider whether the Bell model can be applied to bosons.

Natural critical phenomena are characterized by laminar periods separated by events where bursts of activity take place, and by the interrelated self-similarity of space-time scales and of the event sizes. One example are earthquakes: for this case a new approach to quantify correlations between events reveals new phenomenology. By linking correlated earthquakes one creates a scale-free network of events, which can have applications in hazard assessment. Solar flares are another example of critical phenomenon, where event sizes and time scales are part of a single self-similar scenario: rescaling time by the rate of events with intensity greater than an intensity threshold, the waiting time distributions conform to scaling functions that are independent of the threshold. The concept of self-organized criticality (SOC) is suitable to describe critical phenomena, but we highlight problems with most of the classical models of SOC (usually called sandpiles) to fully capture the space-time complexity of real systems. In order to fix this shortcoming, we put forward a strategy giving good results when applied to the simplest sandpile models.