Phase-Driven Precision Boost in Quantum Compression for Postselected Metrology

TL;DR

This work identifies the noncyclic Pancharatnam phase, originating from coherent system–meter interactions, as a geometric criterion that governs the efficiency of quantum compression channels in postselected metrology. By formulating the channel operator $\hat{\mathds{K}}{(\lambda,\Theta)}$ and decomposing the quantum Fisher information into total and parallel components, the authors show that tuning the postselection parameter $\Theta$ to enforce $\mathcal{Q}^{\parallel}=0$ yields lossless compression and maximal information per trial. The approach is developed for general meter operators and extended to qudit meters, demonstrating that higher-dimensional meters provide substantial gains in $\mathcal{I}^{\perp}$, $\mathcal{T}$, and SNR, with Heisenberg-limited scaling under favorable conditions. The Pancharatnam phase thus serves as a geometric benchmark for designing high-precision quantum parameter estimation protocols in postselected metrology, offering clear strategy for phase control and meter-state engineering. Overall, the paper bridges geometric phase concepts with practical metrology, outlining concrete pathways to surpass classical limits via lossless compression and qudit-enabled enhancements.

Abstract

We reveal the noncyclic Pancharatnam phase--arising from the coherent system-meter interaction--as a fundamental criterion that governs the optimal performance of quantum compression channels in postselected metrology. This phase embodies a geometric connection that enables precise control over the parallel evolution of the meter state, thereby maximizing the quantum Fisher information per trial and achieving lossless compression channels. Remarkably, fine-tuning the postselection parameter just below this optimal phase incurs substantial information loss, whereas tuning it just above fully suppresses undesired parallel evolution, enhancing information retention beyond that achievable in postselected protocols lacking Pancharatnam phase effects. We further reveal that leveraging qudit meter states can unlock a substantial additional enhancement. These findings establish the Pancharatnam phase as a geometric benchmark, guiding the design of high-precision quantum parameter estimation protocols.

Figures (8)

• Figure 1: Postselection probability as a function of the postselection parameter $\Theta$, for $\lambda=10^{-3}rad$, $d=21,n=50$. This produces a Pancharatnam phase shift $\textrm{Im}\ln\langle\hat{\mathcal{O}}_{\lambda}^{P}\rangle=\frac{(d-1)n\lambda}{2}=0.5$. (a) For different values of $j$ with preselected state $m_{i}=j,$ and postselected state $m_{f}=-j$. (b) For $j=1$, $m_{i}=j$ and different values of $m_{f}$.
• Figure 2: Contour plots of the QFI per trial $\mathcal{T}{ (\lambda,\Theta)}$ depicted as functions of the coupling and postselection parameters $\lambda$ and $\Theta$, for quantum numbers $j=1/2$, $1$, and $3/2$, $n=1$, $d=30$. The top row corresponds to calculations using the meter operator $\hat{O}_{\lambda}^{P},$ while the bottom row displays results obtained with $\hat{O}_{\lambda}^{(0)}$. In each panel, the color scale encodes the $\mathcal{T}{ (\lambda,\Theta)}$ magnitude, with lighter shades indicating higher sensitivity. Further discussion and detailed physical interpretation are provided in the main text.
• Figure 3: The postselected QFI $\mathcal{I}^{\perp}$ as a function of $\Theta$ for $n\lambda=10^{-3}$ and $d=30$. (a) QFI exhibits a linear growth as a function of $j$. Notably, the QFI peak shifts towards the Pancharatnam phase as $j$ increases. (b) Reference protocol $\hat{\mathcal{O}}_{\lambda}^{(0)}$ shows qualitative differences in scaling and suppression of QFI as $j$ increases.
• Figure 4: Postselected QFI $\mathcal{I}^{\perp}$, $\mathcal{I}^{T}$, and $\mathcal{I}^{\parallel}$ as a function of $\Theta$ for $n\lambda=10^{-3}$, $d=2$ and arbitrary $j$. (a) For the meter operator $\hat{\mathcal{O}}_{\lambda}^{P}$, the characteristic trade-off between $\mathcal{I}^{\perp}$ and $\mathcal{I}^{\parallel}$ occurs in the vicinity of the Pancharatnam phase $\frac{n\lambda}{2}$. (b) For the meter operator $\hat{\mathcal{O}}_{\lambda}^{(0)}$, both $\mathcal{I}^{\parallel}$ and $\mathcal{I}^{T}$ exhibit cooperative behavior, whereas the postselected QFI $\mathcal{I}^{\perp}$ suffers an approximately 83% reduction at $\Theta_{\perp}=\pm\frac{n\lambda}{2}$, indicating suboptimal performance of operator $\hat{\mathcal{O}}_{\lambda}^{(0)}$ in comparison with $\hat{\mathcal{O}}_{\lambda}^{P}$.
• Figure 5: (a) Variation of the postselected QFI $\mathcal{I}^{\perp}$, $\mathcal{I}^{T}$, and $\mathcal{I}^{\parallel}$ as functions of $\Theta$ for $d=30$, $n\lambda=10^{-3}$ and arbitrary $j$, illustrating how the operator $\hat{\mathcal{O}}_{\lambda}^{P}$ enables precise tuning of the trade-off between orthogonal and parallel sensitivities. (b) The QFI $\mathcal{I}^{\perp}$ plotted against $\Theta$ for various meter dimensions $d$, demonstrating a pronounced increase in QFI and the formation of sharply localized peaks with rising $d$ when using $\hat{\mathcal{O}}_{\lambda}^{P}$. These peaks signify optimal $\Theta$ at which maximal quantum sensitivity is achieved as meter complexity grows. (c) In contrast, the QFI $\mathcal{I}^{\perp}$ corresponding to $\hat{\mathcal{O}}_{\lambda}^{(0)}$ remains essentially unchanged in its maximal numerical value as $d$ increases, reflecting a diminished sensitivity to the meter dimension $d$.