inequalities for the polytope can have physical interpretation: for instance, the Pauli principle is simply one linear inequality bounding the polytope of a fermionic system (see [@coleman63; @borlanddennis72; @ruskai07; @klyachkoaltunbulak08; @klyachko09] and [@higuchi03; @higuchisudberyszulc03; @bravyi04] for further examples).

Random states play a fundamental role in physics. In classical *statistical mechanics*, the canonical state (or, “canonical ensemble”) is the marginal probability distribution (Gibbs measure, or Boltzmann distribution) of states of the system arising from a uniformly random configuration of system and bath, subject to an energy constraint and fixed particle numbers for both system and bath [@huang90]. In quantum statistical mechanics, the canonical state of the system is the reduced density matrix of the uniform state on a subspace (encoding the energy constraint) of system and bath. In fact it can be shown that the canonical state almost always approximates the reduced density matrix of a random pure state in this subspace, as for large systems a concentration of measure occurs [@popescushortwinter06; @lloyd06; @goldsteinlebowitztumulkaetal06]. Earlier, and also out of thermodynamic considerations, Lloyd and Pagels in a seminal work computed the distribution of eigenvalues of the reduced density matrix of the system when system and bath are in a random pure state 