Conditions for Quantum Violation of Macrorealism in Large-spin Limit

TL;DR

This work develops a geometric, information-theoretic framework for entropic Leggett--Garg inequalities (ELGIs) to test macrorealism with higher-order temporal correlations. Using a WKB approach, it shows that in the macroscopic large-spin limit violations for maximally mixed states remain bounded in generic parameter regimes, while breakdowns of the WKB approximation can yield maximal violations at special angular settings. ELGIs are demonstrated to be robust to decoherence and complementary to standard LGIs, and they offer direct probes of non-Markovian memory through entropy-based criteria. The findings clarify the conditions under which macroscopic quantum phenomena can be detected and provide concrete guidance for experiments aiming to observe quantum violations at large scales.

Abstract

This study investigates the emergence of macroscopic classical behavior from quantum foundations via the entropic Leggett--Garg inequality. We introduce a geometric framework for deriving entropic Leggett--Garg inequalities with higher-order temporal correlations and demonstrate their advantages over conventional formulations. Numerical analyses show that entropic Leggett--Garg inequalities offer a robust and complementary criterion to standard approaches, providing a transparent information theoretic interpretation that facilitates the characterization of coherent quantum processes. By applying the WKB approximation, we prove that violations for maximally mixed states remain bounded by a constant in the macroscopic limit, indicating that macrorealism dominates in generic parameter regimes. We further explain previously reported maximal violations at specific parameter regimes as a consequence of the breakdown of the WKB approximation. Our findings indicate that quantum and classical descriptions remain macroscopically incompatible, while violations persist only in fine-tuned regimes, clarifying the conditions for detecting macroscopic quantum phenomena.

Figures (7)

• Figure 1: Schematic representation of the experimental setting for ELGI. The elementary type ELGI $\mathcal{D}_{2,3}=H(Q_{1},Q_{3},Q_{4})+H(Q_{1},Q_{2},Q_{4})-H(Q_{1},Q_{2},Q_{3},Q_{4})-H(Q_{1},Q_{4})\ge0$ is taken as an example, where $Q_{i}$ denotes $Q(t_{i})$ for simplicity. All the $M$ experiments begin with the same initial state $\hat{\rho}_{_{0}}$. The joint entropy must be obtained from independent experiments rather than derived from the marginal probability distribution of a single experiment with measurements at all time points, as later measurements could be affected by earlier ones.
• Figure 2: Comparison of quantum violations of ELGIs involving different orders of temporal correlations. All possible Shannon-type ELGIs are grouped according to their order. ELGIs that can be written as sums of other ELGIs of the same order are excluded, since they do not provide any additional information about quantum violations. Numerical calculations are performed for a spin-2 system with Hamiltonian $\hat{H}=\omega\hat{J}_{y}$ and observable $\hat{Q}=\hat{J}_{z}$, considering (a) $n=3$ and (b) $n=4$ equally spaced measurement time points, $t_{i+1}-t_{i}=\Delta t$. The complete list of inequalities involved in the figure is provided in the supplementary material.
• Figure 3: Comparison of quantum-violation regions for ELGIs, SLGIs, and WLGIs. Shaded regions indicate parameter regimes where at least one inequality in each family is violated: red for ELGIs, green for SLGIs, and blue for WLGIs. Numerical results are obtained for a spin-1 system governed by the Lindblad master equation $\mathrm{d}\hat{\rho}/\mathrm{d}t=-i[\omega \hat{J}_{y},\hat{\rho}]+\gamma(\hat{J}_{-}\hat{\rho}\hat{J}_{+}-\tfrac{1}{2}\{\hat{J}_{+}\hat{J}_{-},\hat{\rho}\})$ with the observable $\hat{Q}=\hat{J}_{z}$ and $n=3$ measurement times. The panels correspond to different decay rates: (a) $\gamma=0.03\omega$ and (b) $\gamma=0.08\omega$, demonstrating that ELGI violations persist under stronger dissipation, whereas violations in the other families may disappear.
• Figure 4: Probing non-Markovianity via ELGI violations. (a) Schematic of the experimental setup for a trapped $^{40}$Ca$^{+}$ qubit with tunable dissipation WZZ+21. The qubit subspace is formed by Zeeman levels $\ket{0}=\ket{4^{2}S_{1/2},m=-1/2}$, $\ket{1}=\ket{3^{2}D_{5/2},m=+1/2}$, and the sink state $\ket{g}=\ket{4^{2}S_{1/2},m=+1/2}$. The $|0\rangle\leftrightarrow|1\rangle$ transition is driven by a 729-nm laser, while a tunable loss from $|1\rangle$ to $|g\rangle$ is induced via the short-lived intermediate state $|e\rangle$ using an 854-nm laser. (b) Numerical ELGI parameter $\mathcal{D}_{1,3}$ for the dissipative qubit system at different dissipation rates $\gamma=0.1,0.3,0.5,0.7$ (as shown in legend), with $n=3$ equally spaced measurement times $t_{i+1}-t_{i}=\Delta t$. Positive violations of the classical Markovianity condition are observed, which gradually weaken as $\gamma$ increases. Dynamical equations and simulation details are provided in the Supplemental Material.
• Figure 5: Comparison of numerical results with asymptotic limits.$n=3$ measurement time points are considered, and the rotation angles are set as $\beta_{1,2}=2\beta$ and $\beta_{2,3}=\beta$, yielding an asymptotic limit of $\mathcal{D}_{2,3}=-\ln{|\sin{3\beta}/\sin{\beta}|}$. Because of the symmetry of the Wigner $d$-matrix, calculations are restricted to $0\le\beta\le\pi$ without loss of generality. Apart from the divergence at $\beta=\pi/3$, where $\beta_{1,3}\to\pi$, the numerical results closely match the asymptotic limit as the spin $j$ increases.