  In the study of quantum limits to parameter estimation, the high
dimensionality of the density operator and that of the unknown parameters have
long been two of the most difficult challenges. Here we propose a theory of
quantum semiparametric estimation that can circumvent both challenges and
produce simple analytic bounds for a class of problems in which the dimensions
are arbitrarily high, few prior assumptions about the density operator are
made, but only a finite number of the unknown parameters are of interest. We
also relate our bounds to Holevo's version of the quantum Cram\'er-Rao bound,
so that they can inherit the asymptotic attainability of the latter in many
cases of interest. The theory is especially relevant to the estimation of a
parameter that can be expressed as a function of the density operator, such as
the expectation value of an observable, the fidelity to a pure state, the
purity, or the von Neumann entropy. Potential applications include quantum
state characterization for many-body systems, optical imaging, and
interferometry, where full tomography of the quantum state is often infeasible
and only a few select properties of the system are of interest.
