The signaling dimension of two-dimensional and polytopic systems

Abstract

The signaling dimension of any given physical system represents its classical simulation cost, that is, the minimum dimension of a classical system capable of reproducing all the input/output correlations of the given system. The signaling dimension landscape is vastly unexplored; the only non-trivial systems whose signaling dimension is known -- other than quantum systems -- are the octahedron and the composition of two squares. Building on previous results by Matsumoto, Kimura, and Frenkel, our first result consists of deriving bounds on the signaling dimension of any system as a function of its Minkowski measure of asymmetry. We use such bounds to prove that the signaling dimension of any two-dimensional system (i.e. with two-dimensional set of admissible states, such as polygons and the real qubit) is two if and only if such a set is centrally symmetric, and three otherwise, thus conclusively settling the problem of the signaling dimension for such systems. Guided by the relevance of symmetries in the two dimensional case, we propose a branch and bound division-free algorithm for the exact computation of the symmetries of any given polytope, in polynomial time in the number of vertices and in factorial time in the dimension of the space. Our second result then consist of providing an algorithm for the exact computation of the signaling dimension of any given system, that outperforms previous proposals by exploiting the aforementioned bounds to improve its pruning techniques and incorporating as a subroutine the aforementioned symmetries-finding algorithm. We apply our algorithm to compute the exact value of the signaling dimension for all rational Platonic, Archimedean, and Catalan solids, and for the class of hyper-octahedral systems up to dimension five.