  The lower bound of estimated accuracy for a parameter unitarily encoded in
closed systems has been obtained, and both optimal initial states and detection
operators can be designed guided by the lower bound. In this letter, we
demonstrate that the lower bound of estimated accuracy for a dissipative
parameter non-unitarily encoded in open systems based on the non-hermitian
linear response theory. This lower bound is related to the correlation of the
encoding dissipative operator for open systems in contrast to the fluctuation
of the encoding operator for closed systems. We also explicitly calculate the
estimated accuracy for dissipative parameters corresponding to three different
kinds of non-unitarily encoding processes, including particle loss, relaxation,
and dephasing, which further confirm this lower bound. We finally compare the
lower bound with the quantum Fisher information obtained by tomography, and we
find they are consistent under suitable initial states and detection operators.
This lower bound can guide us to find the optimal initial states and detection
operators to significantly simplify the measurement process.
