Quantum Cramer-Rao bound for a Massless Scalar Field in de Sitter Space

TL;DR

This paper investigates the quantum limits of estimating cosmological parameters from a massless scalar field in de Sitter space, analyzing quantum Fisher information and measurement bounds during cosmic evolution.

Contribution

It derives explicit quantum Fisher information expressions for a scalar field in de Sitter space and examines how quantum-to-classical transition impacts measurement precision.

Findings

Quantum Fisher information varies during cosmic evolution.

Quantum-to-classical transition affects measurement bounds.

Residual quantum correlations influence estimation precision.

Abstract

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.