  Bayesian estimation of a mixed quantum state can be approximated via maximum
likelihood (MaxLike) estimation when the likelihood function is sharp around
its maximum. Such approximations rely on asymptotic expansions of
multi-dimensional Laplace integrals. When this maximum is on the boundary of
the integration domain, as it is the case when the MaxLike quantum state is not
full rank, such expansions are not standard. We provide here such expansions,
even when this maximum does not belong to the smooth part of the boundary, as
it is the case when the rank deficiency exceeds two. These expansions provide,
aside the MaxLike estimate of the quantum state, confidence intervals for any
observable. They confirm the formula proposed and used without precise
mathematical justifications by the authors in an article recently published in
Physical Review A.
