How Gaussian can the Sky be? Primordial Non-Gaussianity from Quantum
  Information
  Using the quantum information picture to describe the early universe as a
time dependent quantum density matrix, with time playing the role of a
stochastic variable, we compute the non-gaussian features in the distribution
of primordial fluctuations. We use a quasi de Sitter model to compute the
corresponding quantum Fisher information function as the second derivative of
the relative entanglement entropy for the density matrix at two different
times. We define the curvature fluctuations in terms of the time quantum
estimator. Using standard quantum estimation theory we compute the non-gaussian
features in the statistical distribution of primordial fluctuations. Our
approach is model independent and only relies on the existence of a quasi de
Sitter phase. We show that there are primordial non-gaussianities, both in the
form of squeezed and equilateral shapes. The squeezed limit gives a value of
$f_{\rm NL} \sim n_s-1$. In the equilateral limit we find that $f_{\rm NL} \sim
0.03$. The equilateral non-gaussianity is due to the non-linearity of
Einstein's equation. On the other hand, the squeezed one is due to the quantum
nature of clock synchronization and thus real and cannot be gauged away as a
global curvature. We identify a new effect: {\it clock bias} which is a pure
quantum effect and introduces a bias in the spectral tilt and running of the
power spectrum of order $\sim 10^{-4}$, which could be potentially measurable
and yield precious information on the quantum nature of the early Universe.
