论文标题
只有经典的参数化状态在最小二乘损失下具有最佳测量
Only Classical Parameterised States have Optimal Measurements under Least Squares Loss
论文作者
论文摘要
量子状态的测量构成量子信息处理中的关键组成部分。因此，比较测量并确定测量策略是否最佳是一项重要的任务。诸如量子渔民信息之类的熵数量，可捕获渐近最优性，而不是使用有限资源的最佳性。我们介绍了一个框架，该框架可以最终确定在非反应方案中测量是否最佳。我们的方法取决于估计器的预期错误（称为风险）的基本属性，并且不涉及对熵数量的优化。该框架适用于有限的样本量和缺乏先验知识，以及渐近和贝叶斯环境。我们证明了一个无关定理，该定理表明，只有经典状态在最常见的误差测量选择下接受最佳测量：最小二乘。我们进一步考虑了大约最佳测量的限制性较小的概念，并提供了足够的条件，以使这种测量存在。最后，我们概括了估计器何时不可接受的概念（即严格比替代方案更糟），并提供了两个足够的条件，以使测量不可接受。
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.
