Quantum optimality of photon counting for temperature measurement of thermal astronomical sources

Using the quantum Cram\'{e}r-Rao bound from quantum estimation theory, we derive a fundamental quantum limit on the sensitivity of a temperature measurement of a thermal astronomical source. This limit is expressed in terms of the source temperature $T_s$, input spectral bandwidth $\Delta \nu$, and measurement duration $T$, subject to a long measurement time assumption $T\Delta \nu \gg 1$. It is valid for any measurement procedure that yields an unbiased estimate of the source temperature. The limit agrees with the sensitivity of direct detection or photon counting, and also with that of the ideal radiometer in the regime $kT_s/h \nu_0\gg 1$ for which the Rayleigh-Jeans approximation is valid, where $\nu_0$ is the center frequency at which the radiometer operates. While valid across the electromagnetic spectrum, the limit is especially relevant for radio astronomy in this regime, since it implies that no ingenious design or technological improvement can beat an ideal radiometer for temperature measurement. In this connection, our result refutes the recent claim of a radio astronomy technique with much-improved sensitivity over the radiometer (Lieu et al. 2015).