Entropic Uncertainty Relations with Quantum Memory in Accelerated Frames via Unruh-DeWitt Detectors

Abstract: Quantum uncertainty is deeply linked to quantum correlations and relativistic motion. The entropic uncertainty relation with quantum memory offers a powerful way to study how shared entanglement affects measurement precision. However, under acceleration, the Unruh effect can degrade quantum correlations, raising questions about the reliability of QMA-EUR in such settings. Here, we investigate the QMA-EUR for two uniformly accelerating Unruh-DeWitt detectors coupled to a massless scalar field. Using the Kossakowski-Lindblad master equation, we calculate the entropic uncertainty, its lower bound, and the tightness of the relation under different Unruh temperatures. We find that acceleration does not always increase the lower bound on the uncertainty relation. Depending on the initial correlations between the detectors, it may either increase or decrease. This behavior results from the interplay between quantum discord and minimal missing information. Interestingly, a higher quantum discord does not necessarily lead to lower uncertainty.

• The paper introduces an analysis of quantum-memory-assisted entropic uncertainty relations using accelerating UDW detectors to reveal how the Unruh effect increases measurement uncertainty.
• It employs a Kossakowski–Lindblad master equation and numerical simulations to uncover a nontrivial interplay between quantum discord and minimal missing information.
• The findings indicate that relativistic noise fundamentally impacts uncertainty bounds, with significant implications for designing quantum cryptography and information protocols.

Entropic Uncertainty Relations with Quantum Memory under Acceleration: Analysis via Unruh-DeWitt Detectors

Overview and Motivation

This paper conducts a rigorous investigation of quantum-memory-assisted entropic uncertainty relations (QMA-EUR) for systems subjected to relativistic acceleration, specifically focusing on uniformly accelerating Unruh-DeWitt (UDW) detectors coupled to a massless scalar field. The principal aim is to dissect how the Unruh effect—in which an accelerating observer perceives the vacuum as a thermal bath—modulates uncertainty bounds when quantum memory is present. Attention is directed toward the nontrivial interplay between quantum discord (QD) and minimal missing information (MMI), and their joint influence on the practical tightness and reliability of entropic uncertainty bounds in relativistic environments.

Theoretical Framework

A historical context for the EUR is provided, progressing from Heisenberg's original uncertainty principle (variance-based) to entropic measures as formulated by Deutsch and Maassen-Uffink, culminating in the conditional entropy-based QMA-EUR introduced by Berta et al. For a bipartite state $\rho_{AB}$, the standard form is: $$S(X|B) + S(Z|B) \geq \log_2 \left(\frac{1}{c}\right) + S(A|B)$$ where $c$ captures the complementarity of the measurement bases, and $S(A|B)$ can attain negative values for entangled states—a crucial aspect for quantum key distribution. The inequality's tightness is quantified by $\delta = U - \mathcal{B}$.

The focal scenario involves two-level UDW detectors accelerating uniformly through Minkowski spacetime and interacting weakly ($\mu \ll 1$) with a scalar field. Detector evolution is governed by a Kossakowski–Lindblad master equation, leveraging Markovianity due to weak coupling. The dynamics and stationary states are determined analytically, with the initial state parameter $\Delta_0$ encapsulating the two-qubit correlations, spanning singlet ($\Delta_0 = -3$) to triplet ($\Delta_0=1$) regimes.

Numerical Analysis and Key Findings

Numerical evaluation of $U$, $\mathcal{B}$, and $\delta$ is performed for measurements in mutually unbiased bases ($\sigma_x$ and $\sigma_z$), and investigated as a function of the Unruh temperature $T$—the effective temperature experienced due to acceleration.

Strong numerical findings include:

• Monotonic Increase in Uncertainty: Across all cases, the entropy uncertainty $U$ increases as a function of $T$, reflecting degradation of fidelity in measurement outcomes.
• Non-monotonic Tightness: The bound $\mathcal{B}$ and tightness $\delta$ manifest nontrivial dependencies on both $T$ and the initial correlation parameter $\Delta_0$:
  • For $\Delta_0 = -1$, the relation becomes increasingly tight as $T$ increases.
  • For intermediate $\Delta_0$ ($0.5$), tightness first increases then decreases.
  • For highest $\Delta_0$ ($1$), the bound is saturated for $T=0$, but grows less tight with acceleration.
• Discord vs. Missing Information: Crucially, higher quantum discord $D$ does not guarantee a lower uncertainty; the paper demonstrates that the minimal missing information $M$ may override the effects of $D$ (see Eq. $\mathcal{B} = \log_2 \frac{1}{c} + M - D$). When $M$ increases more rapidly than $D$, uncertainty worsens despite potentially stronger quantum correlations.

Implications and Future Directions

The theoretical outcome underscores that the reliability of QMA-EUR in accelerated frame scenarios is not universally predictable from quantum correlation measures alone. The competition between $D$ and $M$ means that both entanglement and classicality, mediated by relativistic noise (Unruh effect), fundamentally affect protocol performance in quantum information tasks. This sensitivity must be accounted for in any practical design of relativistic quantum cryptography, teleportation, or entanglement harvesting wherein detectors are non-inertial.

Additionally, the necessity of the Markovian approximation for analytics signifies a limitation. At extreme accelerations, non-Markovian regimes become relevant as the Unruh time scale approaches the detector’s relaxation times, challenging the validity of the employed master equation. Extension to non-Markovian dynamics is presented as a future research direction.

Conclusion

This work presents a systematic study of QMA-EUR for accelerating Unruh-DeWitt detectors and demonstrates that acceleration-induced thermalization modulates uncertainty bounds in a non-uniform, correlation-dependent manner. The explicit identification of the interplay between quantum discord and minimal missing information imparts clarity to the observed phenomena, cautioning against simplistic interpretations of uncertainty reduction in relativistic quantum protocols. The analysis motivates future investigations into non-Markovian corrections and their impact on the robustness of quantum information in relativistic regimes (2512.10210).