de Sitter空间中无质量标量场的量子Cramer-Rao界
Quantum Cramer-Rao bound for a Massless Scalar Field in de Sitter Space
摘要：
通过对宇宙学量子态进行量子测量，我们能精确地估计宇宙学参数吗？在量子估计理论中，无偏参数估计量的方差从下到上由依赖于测量的Fisher信息的倒数和最终由量子费舍尔信息限定，量子Fisher信息是前者在所有正算子值测量上的最大化。这样的界限被称为量子克拉美-拉奥界限。研究了具有Bunch-Davies真空的无质量标量场在空间平坦FLRW时空中的演化，对于每一个场波数模，都存在双模压缩真空外态。我们得到了量子费舍尔信息以及与占据数测量和功率谱测量相关的费歇尔信息的表达式，并以纯de Sitter展开和de Sitter展开后有辐射支配相位为例，给出了它们演化的具体结果.我们将从宇宙学扰动的量子-经典跃迁的角度讨论这些结果，并定量地说明这种跃迁和剩余量子关联如何影响精度上的界。
英文摘要：
How precisely can we estimate cosmological parameters by performing a quantum measurement on a cosmological quantum state? In quantum estimation theory the variance of an unbiased parameter estimator is bounded from below by the inverse of measurement-dependent Fisher information and ultimately by quantum Fisher information, which is the maximization of the former over all positive operator valued measurements. Such bound is known as the quantum Cramer-Rao bound. We consider the evolution of a massless scalar field with Bunch-Davies vacuum in a spatially flat FLRW spacetime, which results in a two-mode squeezed vacuum out-state for each field wave number mode. We obtain the expressions of the quantum Fisher information as well as the Fisher informations associated to occupation number measurement and power spectrum measurement, and show the specific results of their evoluation for pure de Sitter expansion and de Sitter expansion followed by a radiation-dominated phase as examples. We will discuss these results from the point of view of the quantum-to-classical transition of cosmological perturbations and show quantitatively how this transition and the residual quantum correlations affect the bound on the precision.
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