ABSTRACT
Abstract A closely coupled pair of conjectures/questions-one in differential geometry (by M. Gromov), the other in quantum information theory-are both answered in the negative. The answer derives from a certain metrical flexibility of manifolds and a corresponding improvement to the theoretical efficiency of existing local quantum codes. We exhibit this effect by constructing a family of metrics on S2 × S1, and other three and four dimensional manifolds. Quantitatively, the explicit “freedom” exhibited is too weak (a log1/2 factor in the natural scaling) to yield practical codes but we cannot rule out the possibility of other families of geometries with more dramatic freedom.
