Using a single thermometer to estimate two temperatures

We consider the question: Is it possible to measure two temperatures simultaneously using a single quantum thermometer? In a scenario involving local thermometry, we affirm that this task can indeed be accomplished with the assistance of quantum control. In particular, we consider a composite particle with at least two quantum degrees of freedom (DoF) as a thermometer, where one DoF is susceptible to the local temperature whereas the other DoF is quantum-controlled. For thermalization to two temperatures, the thermometer is exposed to two baths at distinct temperatures with a quantum controlled interaction, or a purified single bath whose state is quantum controlled in addition to the interaction. We show that such a particle, if used in a Mach-Zehnder interferometer or a quantum switch with such engineered bath(s)/interactions, can be used to estimate two temperatures simultaneously. For all of the setups -- that allow simultaneous two temperature thermometry -- which we consider, we obtain the variance in the estimated temperatures through the multi-parameter Cram'er-Rao bound. Our results establish the utility of quantum control in a novel metrological task besides opening new avenues for further applications, such as quantum thermodynamics with two thermal baths along with quantum control.