Metrological approach to the emergence of classical objectivity

TL;DR

This paper recasts the emergence of classical objectivity within quantum Darwinism as an operational metrology problem by using quantum Fisher information (QFI) to quantify how precisely an observer can infer the system from environmental fragments. In a spin-star model, it derives explicit timescales: the decoherence time $\tau_D=1/\sqrt{2\langle J^2\rangle}$, the fragment-imprinting time $\tau_F=1/(2\sqrt{f\langle J^2\rangle})$, and the local-measurement timescale $\tau_Y=(2|\sin(2\theta)\langle J\rangle|\sqrt{f})^{-1}$, showing that the maximal QFI saturates at 4 and that QFI grows exponentially with time for optimal measurements. The results reveal that while optimal measurements on environmental fragments yield the fastest emergence of objectivity, suboptimal (static local) measurements can still saturate the Cramér-Rao bound in the thermodynamic limit given sufficient time, thereby supporting the robustness of quantum Darwinism as a mechanism for classical objectivity. The work also provides numerical evidence of redundancy-like behavior (plateaus in information gain) for mesoscopic environments and discusses extensions to more complex environments and multiparameter scenarios.

Abstract

We present a precise characterization of the onset of classicality that combines the formalism of quantum Darwinism with the tools from quantum metrology. We show that the quantum Fisher information provides a useful metric for assessing the rate at which classical objectivity emerges. Furthermore, our formalism allows us to explore how the choice of measurement impacts the precision with which an observer can determine the state of the system. For a paradigmatic example of the spin-star model, we demonstrate that optimal measurements lead to the emergence of classicality at an exponential rate. Although other measurements necessarily lead to slower emergence, we importantly show that suboptimal measurements can still saturate the Cramér-Rao bound. By recasting emergent classicality as an information acquisition protocol, our framework provides a precise operational description of quantum Darwinism.

Figures (4)

• Figure 1: (a) Diagram of the spin-star model for $N=6$ and $f=1/3$. The environmental fragment accessible to an observer is shown in the red region and the remaining environment in the green region. (b) Thermodynamic limit of QFI (red, solid line) against time. Also shown is the corresponding precision for $S_y$ with $\tau_Y/\tau_F=\{1,2,5\}$ (blue dashed, green dotted and black dot-dashed lines).
• Figure 2: QFI, $\mathcal{F}_\theta$, and precision, $1/\text{Var}\left(\theta\right)$, against time $t$ for $10$ different realisations of the coupling strengths $J$ (blue semitransparent lines). The asymptotic behaviour, i.e. thermodynamic limit, is also shown by the red, dashed line. We choose $f=0.2$, $\mathcal{J}=0.5$ and $\sigma=0.5$. Different columns corresponds to increasing system size $N=25$ (left) and $N=50$ (right). Dashed vertical grey lines at $t=\tau_{F,Y}$ and $t=\sqrt{N}$.
• Figure 3: QFI against fragment fraction $f$ and time $t$ for (a) a single realizations of coupling strengths for $\mathcal{J}=0.5$, $\sigma=0.5$ and $N=30$ (b) thermodynamic limit. Black dashed lines show $t=\tau_F$ and $t=\sqrt{N}$.
• Figure 4: The behaviour of (a) QFI and (b) precision with $S_y$ measurement with increasing fragment fraction $f$ at a late time $t=3$ with $N=30$, $\mathcal{J}=0.5$ and $\sigma=0.5$. Dashed blue line shows the average behaviour from $2000$ random realisations of the coupling. Realisations within one standard deviation of the mean reside in the blue shaded area bordered with green lines and red dots show the thermodynamic limit.