  Measurements of quantum states form a key component in quantum-information
processing. It is therefore an important task to compare measurements and
furthermore decide if a measurement strategy is optimal. Entropic quantities,
such as the quantum Fisher information, capture asymptotic optimality but not
optimality with finite resources. We introduce a framework that allows one to
conclusively establish if a measurement is optimal in the non-asymptotic
regime. Our method relies on the fundamental property of expected errors of
estimators, known as risk, and it does not involve optimisation over entropic
quantities. The framework applies to finite sample sizes and lack of prior
knowledge, as well as to the asymptotic and Bayesian settings. We prove a no-go
theorem that shows that only classical states admit optimal measurements under
the most common choice of error measurement: least squares. We further consider
the less restrictive notion of an approximately optimal measurement and give
sufficient conditions for such measurements to exist. Finally, we generalise
the notion of when an estimator is inadmissible (i.e. strictly worse than an
alternative), and provide two sufficient conditions for a measurement to be
inadmissible.
