  Improving the measurement precision of temperature is very important and
challenging, especially in the low temperature range. Based on the existence of
invariant subspaces during the polariton thermalization, we propose a new way
to enhance the measurement precision of the low temperature and obtain Landau
bound to avoid that the measurement uncertainty of the temperature diverges as
the temperature approaches zero. The measurement precision of the low
temperature increases significantly with the number of polariton states. In
order to resist the dissipation, the incoherent pumping is necessary for
obtaining the information of the temperature encoded in the steady state. It
should be noted that too strong incoherent pumping is wasteful due to that the
quantum Fisher information of the temperature becomes less and less dependent
on the total number of the polaritons.
