In the paper "Continuous-Variable Quantum Key Distribution with Gaussian Modulation – The Theory of Practical Implementations" by Fabian Laudenbach, there's a distinction between the modulation variance of the quadrature components $q$ and $p$, $\tilde{V}_{mod}$; and the modulation variance of the quadrature operators $\hat{q}$ and $\hat{p}$, $V_{mod}$. The relation between them is given as $V_{mod} = 4 \tilde{V}_{mod}$.
In the context of parameter estimation, it seems that the ${V}_{mod}$ used in the formulas is the modulation variance of the quadrature operators, not the quadrature components.
I'm currently developing a simulation code, and I'm unsure whether to generate Alice's coherent states using a Gaussian distribution of the form $N(0,V_{mod})$ or $N(0,\tilde{V}_{mod})$. Which one is the correct one to use in this case?
What is the difference between both variance?
