Fundamental limits on low-temperature quantum thermometry with finite resolution

Patrick P. Potts, Jonatan Bohr Brask, and Nicolas Brunner

Department of Applied Physics, University of Geneva, Switzerland

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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 tight-binding 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 square-lattice Ising model, highlighting the role of phase transitions.

The precise measurement of low temperatures at small length scales is of great interest for modern technologies, promising benefits for applications in many fields ranging from quantum information to molecular biology. However, fundamental limitations on the precision of such measurements are not yet fully understood. Here we develop a general approach to low-temperature quantum thermometry which allows for taking into account the restrictions imposed by the measurement process. We find that the restriction of a finite resolution, a restriction which is present in most experiments, results in a fundamental bound on the precision of any temperature measurement.

Interestingly, this fundamental bound allows, at least in principle, for infinitely precise measurements as temperature approaches zero. This is in stark contrast to the exponential divergence of the measurement error, which dominates many low-temperature thermometry strategies. Our approach provides an intuitive understanding of the mechanism that allows overcoming this exponential scaling as well as a visual tool to easily identify such scenarios. Through these insights, we were able to demonstrate that measurements with finite resolution can indeed result in a polynomially diverging measurement error in an experimentally relevant physical setup. Our approach thus provides a promising route for identifying systems favorable for low-temperature quantum thermometry, paving the way for temperature measurements with a precision that is only limited by fundamental constraints.

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