Quantumness and entropic uncertainty for a pair of static Unruh-DeWitt detectors

In this study, we investigate a pair of detectors operating in Minkowski space-time and analyze the characteristics of various quantum resources within this framework. Specifically, we focus on examining the properties of Bell nonlocality, quantum coherence, the nonlocal advantage of quantum coherence (NAQC), and measured uncertainty in relation to the energy ratio and the distance between the detectors. Additionally, we examine how the initial states influence these quantum properties. Notably, our findings reveal that both a larger energy ratio and a greater separation between the detectors degrade the system’s quantumness. Moreover, we explore the evolution of entropic uncertainty and demonstrate its inverse correlation with both Bell nonlocality and coherence, highlighting the intricate interplay between these quantum resources. These insights provide a deeper understanding of quantumness in a relativistic framework and may contribute to the ongoing discussion on the black hole information paradox.

💡 Research Summary

This paper investigates the quantum resources of a pair of static Unruh‑DeWitt detectors placed in (3+1)‑dimensional Minkowski spacetime. The detectors are modeled as identical two‑level systems with the same energy gap ΔE and coupling constant ν, initially prepared in the entangled state |ϕ⟩ = sinθ|00⟩ + cosθ|11⟩. Their interaction with a real scalar vacuum field is described by a monopole coupling, and the dynamics are treated to second order in perturbation theory, yielding an evolved two‑qubit density matrix ρ_AB(t). The key parameters governing the dynamics are the energy‑ratio ΔF = (ΔE² – (mc²)²)/(mc²)², which measures the detector‑field energy mismatch, and the spatial separation d between the detectors.

Four quantum resources are examined: (i) Bell nonlocality, quantified by the maximal CHSH violation B_max = 2√M_AB where M_AB is derived from the correlation matrix of ρ_AB; (ii) l₁‑norm coherence C_{l1} = Σ_{i≠j}|ρ_{ij}|; (iii) relative‑entropy coherence C_{REC} = S(ρ_diag) – S(ρ); (iv) the nonlocal advantage of quantum coherence (NAQC), defined as the excess of average post‑measurement coherence over the single‑qubit coherence bound, evaluated for both l₁‑norm and REC. Additionally, an entropic uncertainty relation based on the sum of two Maassen‑Uffink entropies is used to assess measurement uncertainty.

Numerical analysis shows that Bell nonlocality exhibits a bell‑shaped dependence on the state parameter θ, reaching its peak at θ = π/4. As ΔF increases, the region of θ that yields a CHSH violation shrinks, indicating that a larger detector‑field energy mismatch suppresses nonlocal correlations. Both coherence measures display the same qualitative behavior: they increase from zero, peak at θ = π/4, and then decline; higher ΔF reduces their maximal values. The distance d also degrades coherence, but even for d → ∞ the coherence saturates at a non‑zero value, reflecting the persistence of certain nonlocal features independent of spatial separation.

NAQC follows the same trends as Bell nonlocality and coherence. For small ΔF and short distances, NAQC is maximal, while it diminishes as either parameter grows. The study further reveals an inverse relationship between entropic uncertainty and the other resources: when B_max·C_{l1} is large, the uncertainty is reduced, whereas larger ΔF or d increase the uncertainty. This confirms that abundant quantum resources lead to more predictable measurement outcomes.

Overall, the work demonstrates that in a relativistic setting the energy ratio and inter‑detector distance act as tunable knobs that simultaneously control Bell nonlocality, coherence, NAQC, and measurement uncertainty. The findings have implications for relativistic quantum information tasks, such as quantum communication near black‑hole horizons, where static detectors in Minkowski space serve as analogues of freely falling observers. The paper provides a clear, analytically tractable framework for exploring how relativistic effects influence the interplay among various quantum resources.