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

TL;DR

This paper establishes a fundamental quantum limit on the sensitivity of temperature measurements of thermal astronomical sources, showing that photon counting and radiometry are optimal and cannot be surpassed.

Contribution

It derives a quantum Cramér-Rao bound for temperature measurement sensitivity, confirming the optimality of photon counting and radiometry in radio astronomy.

Findings

Photon counting achieves the quantum limit for temperature sensitivity.

The ideal radiometer's sensitivity matches the quantum limit in the Rayleigh-Jeans regime.

The result refutes claims of superior sensitivity techniques in radio astronomy.

Abstract

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).