| Summary: | The conventional formulation of quantum sensing is based on the assumption that the probe is reset to its initial state after each measurement. In a very distinct approach, one can also pursue a sequential measurement scheme in which time-consuming resetting is avoided. In this situation, every measurement outcome effectively comes from a different probe, yet is correlated with other data samples. Finding a proper description for the precision of sequential measurement sensing is very challenging as it requires the analysis of long sequences with exponentially large outcomes. Here, we develop a recursive formula and an efficient Monte Carlo approach to calculate the Fisher information, as a figure of merit for sensing precision, for arbitrary lengths of sequential measurements. Our results show that the value of the Fisher information initially increases nonlinearly with the number of measurements and then asymptotically saturates to a linear scaling. This transition, which fundamentally constrains the extractable information about the parameter of interest, is directly linked to the finite memory of the probe when it undergoes multiple sequential measurements. Based on these findings, we establish a figure of merit to determine the optimal measurement sequence length and exemplify our results in three different physical systems.
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