  The weak value amplification technique has been proved useful for precision
metrology in both theory and experiment. To explore the ultimate performance of
weak value amplification for multi-parameter estimation, we investigate a
general weak measurement formalism with assistance of high-order
Hermite-Gaussian pointer and quantum Fisher information matrix. Theoretical
analysis shows that the ultimate precision of our scheme is improved by a
factor of square root of 2n+1, where n is the order of Hermite-Gaussian mode.
Moreover, the parameters' estimation precision can approach the precision limit
with maximum likelihood estimation method and homodyne method. We have also
given a proof-of-principle experimental setup to validate the H-G pointer
theory and explore its potential applications in precision metrology.
