  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.
