Comparing physical quantities with finite-precision: beyond standard metrology and an illustration for cooling in quantum processes

TL;DR

This work tackles the problem of comparing physical quantities when estimates are inherently finite-precision, yielding patches rather than exact values. It introduces a percentile-based uncertainty framework built on minimum-variance unbiased estimators, quantum Fisher information, and a maximum-entropy reconstruction of the estimator distribution, ensuring applicability to asymmetric error profiles. The framework is applied to a three-qubit quantum absorption refrigerator under Markovian dynamics to define and detect finite-precision cooling across transient and steady states in both strong and weak inter-qubit coupling. The results demonstrate that cooling can be identified within finite precision and provide a statistically principled foundation for analyzing finite-precision effects in quantum thermodynamics and metrology. Overall, the approach extends quantum metrology to operational comparisons under realistic measurement constraints and offers a versatile tool for studying finite-precision phenomena in quantum processes.

Abstract

We propose a general framework to compare the values of a physical quantity pertaining to two - or more - physical setups, in the finite-precision scenario. Such a situation requires us to compare between two "patches" on the real line instead of two numbers. Identification of extent of the patches is typically done via standard deviation, as obtained within usual quantum metrological considerations, but can not be always applied, especially for asymmetric error distributions. The extent can however be universally determined by utilizing the concept of percentiles of the probability distribution of the corresponding estimator. As an application, we introduce the concept of finite-precision cooling in a generic quantum system. We use this approach in the working of a three-qubit quantum refrigerator governed by Markovian dynamics, and demonstrate the occurrence of cooling within finite precision for both transient and steady-state regimes, across strong- and weak-coupling limits of the inter-qubit interaction.

Figures (2)

• Figure 1: Finite-precision cooling of the cold qubit in a 3-qubit quantum absorption refrigerator under strong-coupling limit.(a) The temperature $T^f_1(t)$ of the qubit (brown dashed line) is plotted as a function of time $t$, with the steady-state temperature $T^s_1=0.33$ indicated by the red dotted line. Time is shown on the horizontal axis, and temperature on the vertical axis. To examine cooling in the finite-precision sense, we select $9$ representative points along $T^f_1(t)$, spanning from the transient to steady-state regimes. The superscripts $f$ and $s$ denote transient and steady-state values, respectively, as illustrated in the insets. The points are marked in yellow circles and are approximately equally spaced with an interval of $0.05$. The first $8$ points correspond to the transient regime, while the final point represents the steady state. The insets highlight cooling at two points (pink arrows): a transient value $T^t_1(t)=0.50$ (left) and the steady-state temperature $T^s_1=0.33$ (right). For each inset, the horizontal axis shows the percentile range over which cooling occurs, and the vertical axis plots the cooling magnitude $\Delta\bar{T}_i$ for each percentile $i$. Percentile ranges in the inset for $T^s_1$ are labeled alternately along the horizontal axis for clarity. (b) The $3$D plot presents the percentiles $(i,100-i)$, where $i$ is a fixed, positive integer denoting the percentile for which the finite-precision cooling occurs for the first time for each of the $9$ points from panel (a), along with the points and the cooling magnitudes. The horizontal plane shows the percentiles and corresponding data points (rounded up to $2$ decimal places), while the vertical axis represents the cooling magnitudes $\Delta\bar{T}_1$ ( yellow bars). All parameters plotted are dimensionless.
• Figure 2: Illustration of finite-precision cooling of the cold qubit in a 3-qubit quantum absorption refrigerator in the weak-coupling limit.(a) Variation of temperature $T^f_1(t)$ of the qubit (green dashed line), with time $t$. The time is plotted along the horizontal axis, whereas the temperature is plotted along the vertical axis. The blue dotted line is the steady-state temperature of the qubit, $T^s_1=0.36$. To explicitly check for cooling using percentiles, we consider $9$ data points from the curve of the qubit temperature $T^f_1(t)$, across the transient and the steady-state regions. Here, the superscript $f$ is labelled as $t$ (transient) and $s$ (steady state), as illustrated in the inset plots. The points, marked by orange circles, are evenly spaced with an interval of approximately $0.05$. The first $8$ points correspond to the transient regime, while the final point represents the steady-state temperature. The insets highlight the occurrence of cooling for two points (indicated by green arrows): the steady state value $T^s_1=0.36$ (right), and a transient value $T^t_1(t)=0.52$ (left). In each of the inset plots, the percentile range across which the cooling occurs is plotted along the horizontal axis, while the corresponding cooling magnitudes $\Delta\bar{T}_i$, for each value $i$ within the percentile range is plotted along the vertical axis. For better readability, percentile ranges are labelled alternately along the horizontal axis in the inset for $T^s_1$. (b) The $3$D plot depicts the percentiles $(i,100-i)$, where $i$ is a fixed positive integer indicating the point at which cooling, in the finite-precision sense, is first detected for each of the $9$ data points considered in panel (a), along with the data points and the corresponding cooling magnitudes. The percentiles and the data points, rounded to $2$ decimal places, are plotted in the horizontal plane, and the corresponding cooling magnitudes $\Delta\bar{T}_1$ (shown by light red bars) are plotted along the vertical axis. All quantities appearing in the plots are dimensionless.