Quantum tetrahedron and its classical limit

Classical information that is retrieved from a quantum tetrahedron is intrinsically fuzzy. We present an asymptotically optimal generalized measurement for its extraction. For a single tetrahedron the optimal uncertainty in dihedral angles is shown to scale as an inverse of the surface area. The introduced commutative observables allow us to demonstrate how the clustering of many small tetrahedra leads to a faster convergence to a classical geometry.