Fig. 2

Bound on temperature signal-to-noise ratio. The coloured plot shows the optimal signal-to-noise ratio \((T{\mathrm{/\Delta }}T_{\cal S})_{{\mathrm{opt}}}^2\) of an unbiased temperature estimate for the damped oscillator, as a function of temperature T and coupling strength γ. This optimal measurement is determined by the quantum Fisher information, which places an asymptotically achievable lower bound on the temperature fluctuations \({\mathrm{\Delta }}T_{\cal S}\) through the Cramér-Rao inequality¹³. The mesh plot shows the upper bound on \((T{\mathrm{/\Delta }}T_{\cal S})_{{\mathrm{opt}}}^2\) derived here from the generalised thermodynamic uncertainty relation Eq. (16). This uncertainty relation links the temperature fluctuations to the heat capacity of the system at arbitrary coupling strengths. It can be seen that the upper bound becomes tight in both the high temperature and weak coupling limits