In his work on information processing in GPTs http://arxiv.org/abs/quant-ph/0508211 Barrett speculates that the trade-off between allowed states and the allowed dynamics in a GPT is optimal in quantum theory, allowing for information processing capabilities not all of which occur in either the generalized no-signalling theory (GNST) or the generalized local theory (GLT). I'm wondering if this optimality is related to the symmetry of the associated polytope (of allowed states) in the theory, and in what exact manner. Any ideas?
There has indeed been some work on relating the geometry of the state space to the limitations of the theory. First, work relating the local state space to the non-locality present in a theory developed by Janotta et al. They consider the local state space to be a regular polygon. For a large number of sides, the non-locality tends towards Tsirelson's bound. That is, in the infinite limit, the state space is quantum and so, self-dual. The issue of self-duality of state space has been explored and means that the space of effects is isomorphic to the state space.
Relating this to the optimality of quantum state space one can think about reversible computation. In order to have a model of quantum computing that encompasses the circuit model and more general state spaces, one can develop reversible computing for all possible GPTs. Firstly, reversible computations in box-world are trivial as shown by Gross et al. This means that reversible dynamics for box-world is limited to permutations and relabelling of data. If we want reversible dynamics on a less trivial level, e.g. the power to map from one bit to another bit, then this leads to self-duality as shown by Mueller and Ududec. Since the self-duality in the previous paragraph indicates a trade-off in non-locality, they also speculate that this reversible computation limits non-locality. So they connect a computational principle to the structure of the state space, very much in the spirit of Barrett's paper.