Only Classical Parameterised States have Optimal Measurements under Least Squares Loss

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.