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