I am using Kalman filter to estimate the states of a stochastic dynamical system which has very very small noise( consider zero ). The filter is not aware that the noise is zero. Implementation of KF to estimate the states of this dynamical system gives a value, say $\hat{\mathbf{X}}_1\left(t\right)$, at any given time $t$. Let us also assume that we know the true values of the state variables by some method( explanation of the method not important ). Next we manually add some noise to the system and once again apply the KF to estimate the states. Lets call the new states $\hat{\mathbf{X}}_2\left(t\right)$. Let us call the true values $\mathbf{X}\left(t\right)$. Under what conditions the following holds true? ($\|.\|$ indicates 2-norm)
$\|\mathbf{X}\left(t\right) - \hat{\mathbf{X}}_1\left(t\right)\| > \|\mathbf{X}\left(t\right) - \hat{\mathbf{X}}_2\left(t\right)\| $
Short answer is yes, addition of noise can improve the filter's performance, especially in the case of a nonlinear filter. Linearization adds some error, which needs to be accounted for somewhere. Your filter might be overconfident, in which case it would not respond to new measurements and could have errors in the state estimates.