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Precision Enhancement in Transient Quantum Thermometry:Cold-Probe Bias and Its Removal

Debarupa Saha, Ujjwal Sen

TL;DR

This work analyzes how the initial state of a quantum probe affects transient temperature estimation in quantum thermometry. For a qubit thermometer under Markovian dynamics, the authors prove a universal colder-probe bias: transient precision enhancement (as measured by the QFI) is possible if and only if the probe starts colder than the bath, with the criterion $\\beta_i>\\beta$ and independence from the energy gap. By introducing non-Markovianity through an auxiliary mediator, the bias is removed: both hot and cold probes can reach the same maximal transient QFI that surpasses the steady-state limit, albeit with different asymptotic behavior. These results establish a sharp distinction between Markovian and non-Markovian quantum thermometry and offer practical guidance for designing fast, high-precision quantum thermometers, including strategies to engineer non-Markovianity to overcome initial-state biases.

Abstract

We unveil a temperature bias of the probe in transient quantum thermometry under Markovian dynamics. Specifically, for qubit thermometers evolving under Markovian dynamics, we show that enhanced precision beyond the steady state limit can be achieved if and only if the probe is initially colder than the thermal state corresponding to the bath temperature to be estimated. In contrast, this temperature bias can be lifted when the probe dynamics is non-Markovian. In the non-Markovian regime, both hot and cold probes can simultaneously attain the same transient maximum precision, well above the steady-state value.

Precision Enhancement in Transient Quantum Thermometry:Cold-Probe Bias and Its Removal

TL;DR

This work analyzes how the initial state of a quantum probe affects transient temperature estimation in quantum thermometry. For a qubit thermometer under Markovian dynamics, the authors prove a universal colder-probe bias: transient precision enhancement (as measured by the QFI) is possible if and only if the probe starts colder than the bath, with the criterion and independence from the energy gap. By introducing non-Markovianity through an auxiliary mediator, the bias is removed: both hot and cold probes can reach the same maximal transient QFI that surpasses the steady-state limit, albeit with different asymptotic behavior. These results establish a sharp distinction between Markovian and non-Markovian quantum thermometry and offer practical guidance for designing fast, high-precision quantum thermometers, including strategies to engineer non-Markovianity to overcome initial-state biases.

Abstract

We unveil a temperature bias of the probe in transient quantum thermometry under Markovian dynamics. Specifically, for qubit thermometers evolving under Markovian dynamics, we show that enhanced precision beyond the steady state limit can be achieved if and only if the probe is initially colder than the thermal state corresponding to the bath temperature to be estimated. In contrast, this temperature bias can be lifted when the probe dynamics is non-Markovian. In the non-Markovian regime, both hot and cold probes can simultaneously attain the same transient maximum precision, well above the steady-state value.
Paper Structure (8 sections, 1 theorem, 72 equations, 2 figures)

This paper contains 8 sections, 1 theorem, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum Thermometry
  4. The set-up for Markovian dynamics
  5. The set-up for non Markovian dynamics
  6. Colder probe bias under Markovian dynamics
  7. The non-Markovian case and removal of bias
  8. Conclusion and Discussion

Key Result

Theorem 1

Let the initial probe state $\rho_i$, of the form given in Eq. 16, with inverse temperature $\beta_i=\frac{1}{2\omega}\ln\!\left(\frac{1-p}{p}\right)$, evolve under Markovian dynamics while interacting with a thermal bath at inverse temperature $\beta \in(0,\infty)$. Let $\rho_S(\beta,t)$ denote the

Figures (2)

  • Figure 1: Analogy between classical and quantum thermometry. The figure illustrates two scenarios classical thermometry on the left and quantum thermometry on the right. In the classical case both initially cold and initially hot thermometers after interacting with the thermal bath for a sufficiently long time equilibrate with the bath and achieve the same maximal precision in temperature estimation. Thus the long time precision is independent of the initial state of the thermometer. In contrast in the quantum case the probe and bath dynamics leads to a strong dependence of the transient estimation precision on the initial state of the probe. In particular probes prepared at a temperature lower than that of the bath can yield enhanced precision at finite times whereas initially hotter probes never surpass the precision attainable by the thermal state throughout the evolution. This highlights a genuine quantum bias in the choice of the initial probe state for achieving enhanced precision at transient times.
  • Figure 2: Quantum Fisher information along the evolution time. The figure shows the behavior of the QFI corresponding to the inverse temperature, $\beta$, of the bath for two values, $\beta=0.2$ (left) and $\beta=0.5$ (right). In both plots, the red line corresponds to the initially colder probe with $p=0$, while the blue line corresponds to the initially hotter probe with $p=0.5$. The orange dashed line represents the steady-state QFI for the colder initial probe, and the green dashed line represents the steady-state QFI for the hotter initial probe. It can be seen that due to the non-Markovian nature of the probe dynamics, the QFI, $\mathcal{F}(\beta,t)$, exhibits an oscillatory behavior. However, for both hot and cold probes, the maximum achievable precision is always equal and larger than the respective steady-state precision. Remarkably, this equal maximum precision is attained simultaneously at transient times, showing that both hotter and colder initial probes can achieve the same maximum precision.

Theorems & Definitions (2)

  • Theorem 1
  • proof