Fundamental limits on low-temperature quantum thermometry with finite resolution

Abstract

While the ability to measure low temperatures accurately in quantum systems is important in a wide range of experiments, the possibilities and the fundamental limits of quantum thermometry are not yet fully understood theoretically. Here we develop a general approach to low-temperature quantum thermometry, taking into account restrictions arising not only from the sample but also from the measurement process. We derive a fundamental bound on the minimal uncertainty for any temperature measurement that has a finite resolution. A similar bound can be obtained from the third law of thermodynamics. Moreover, we identify a mechanism enabling sub-exponential scaling, even in the regime of finite resolution. We illustrate this effect in the case of thermometry on a fermionic tightbinding chain with access to only two lattice sites, where we find a quadratic divergence of the uncertainty. We also give illustrative examples of ideal quantum gases and a squarelattice Ising model, highlighting the role of phase transitions.

Tight bound on finite-resolution quantum thermometry at low temperatures

“…Our results also demonstrate that thermometry with a vanishing absolute error at low temperature is possible with finite resolution, answering an interesting question left open by previous work [31,34,36]. For systems with a heat capacity that vanishes at low temperatures, a property often included in the third law of thermodynamics, the relative error must diverge, regardless of the available resolution [31]. The absolute error may either also diverge, stay constant, or vanish, with the latter thus being the best behavior one can hope for.…”

“…This results in a measurement-independent quantity, the quantum Fisher information F Q T [40]. Within the canonical ensemble, it can be shown that a projective measurement of the system energy is optimal [31,37]. The quantum Fisher information is then related to the variance of the system energy…”

“…In the probe-sample picture, the size of the probe will limit the amount of information which can be extracted about the sample. More generally, any measurement on the sample, implemented using a finite-sized apparatus, comes with a lower bound on the attainable resolution of, e.g., the system energy spectrum [30][31][32]. Similar restrictions apply in situations where measurements can be made on only part of a large sample [33][34][35], and clearly such finite-resolution constraints must play an important role in formulating fundamental bounds on the attainable thermometric sensitivity.…”

“…We furthermore demonstrate that this scaling can be attained using a single-qubit probe, showing that the bound is tight. To derive our bound, we build upon the framework for finite-resolution quantum thermometry introduced by Potts, Brask, and Brunner in [31].…”

“…In Sec. II, we introduce a general temperature estimation procedure, following [31] and discuss the fundamental precision bounds imposed by the third law of thermodynamics. In Sec.…”