  We investigate the quantum Cramer-Rao bounds on the joint multiple-parameter
estimation with the Gaussian state as a probe. We derive the explicit right
logarithmic derivative and symmetric logarithmic derivative operators in such a
situation. We compute the corresponding quantum Fisher information matrices,
and find that they can be fully expressed in terms of the mean displacement and
covariance matrix of the Gaussian state. Finally, we give some examples to show
the utility of our analytical results.
