量子态估计的Bhattacharyya不等式
Bhattacharyya inequality for quantum state estimation
摘要：
利用参数的高阶导数，给出了量子估计问题中无偏估计的方差下界。这是经典统计估计中Bhattacharyya不等式的量子版本。由于算子乘法的非交换性，我们得到了S型、R型和L型三种不同类型的下界。如果参数是实数，则S型界限很有用。如果参数是复杂的，则R型和L型界限很有用。作为应用，我们将考虑量子高斯态复振幅的多项式估计。对于振幅位于实轴上的情况，将使用S型界导出振幅平方的一致最优估计。它将被证明，有没有无偏估计均匀最优的多项式的湮灭和/或创建运营商的立方振幅。对于振幅不一定位于实轴上的情形，本文导出了振幅的全纯、反全纯和实值多项式的一致最优估计。这些估计的全纯和实值的情况下，达到R型界，而那些反对全纯和实值的情况下，达到L型界。本文阐明了测量激光能量的最佳方法。
英文摘要：
Using higher-order derivative with respect to the parameter, we will give lower bounds for variance of unbiased estimators in quantum estimation problems. This is a quantum version of the Bhattacharyya inequality in the classical statistical estimation. Because of non-commutativity of operator multiplication, we obtain three different types of lower bounds; Type S, Type R and Type L. If the parameter is a real number, the Type S bound is useful. If the parameter is complex, the Type R and L bounds are useful. As an application, we will consider estimation of polynomials of the complex amplitude of the quantum Gaussian state. For the case where the amplitude lies in the real axis, a uniformly optimum estimator for the square of the amplitude will be derived using the Type S bound. It will be shown that there is no unbiased estimator uniformly optimum as a polynomial of annihilation and/or creation operators for the cube of the amplitude. For the case where the amplitude does not necessarily lie in the real axis, uniformly optimum estimators for holomorphic, antiholomorphic and real-valued polynomials of the amplitude will be derived. Those estimators for the holomorphic and real-valued cases attains the Type R bound, and those for the antiholomorphic and real-valued cases attains the Type L bound. This article clarifies what is the best method to measure energy of laser.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
