  It is known that temperature estimates of macroscopic systems in equilibrium
are most precise when their energy fluctuations are large. However, for
nanoscale systems deviations from standard thermodynamics arise due to their
interactions with the environment. Here we include such interactions and, using
quantum estimation theory, derive a generalised thermodynamic uncertainty
relation valid for classical and quantum systems at all coupling strengths. We
show that the non-commutativity between the system's state and its effective
energy operator gives rise to quantum fluctuations that increase the
temperature uncertainty. Surprisingly, these additional fluctuations are
described by the average Wigner-Yanase-Dyson skew information. We demonstrate
that the temperature's signal-to-noise ratio is constrained by the heat
capacity plus a dissipative term arising from the non-negligible interactions.
These findings shed light on the interplay between classical and non-classical
fluctuations in quantum thermodynamics and will inform the design of optimal
nanoscale thermometers.
