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

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.

💡 Research Summary

This paper presents a detailed investigation into the behavior of the Quantum-Memory-Assisted Entropic Uncertainty Relation (QMA-EUR) within a relativistic quantum setting, specifically for two uniformly accelerating Unruh-DeWitt detectors. The QMA-EUR, which quantifies the fundamental limit of predicting the outcomes of two incompatible quantum measurements when the observer shares quantum correlations (memory) with the system, is studied under the influence of the Unruh effect. This effect causes an accelerating observer to perceive the quantum vacuum as a thermal bath at a temperature proportional to its acceleration, potentially degrading quantum correlations.

The authors employ a model where two identical two-level detectors, weakly coupled to a massless scalar field in (3+1)-dimensional Minkowski spacetime, accelerate in unison. The system’s open quantum dynamics, induced by interaction with the field environment, are governed by a Markovian master equation of the Kossakowski-Lindblad type. An analytical expression for the steady-state density matrix of the two-detector system is derived, parameterized by the Unruh temperature (T) and a single parameter (Δ0) that encapsulates the initial quantum correlations between the detectors, ranging from -3 (maximally anti-correlated, like a singlet) to +1 (maximally correlated, like a triplet).

The core of the analysis involves calculating the entropic uncertainty (U = S(X|B) + S(Z|B)) for Pauli X and Z measurements on one detector, its theoretical lower bound (B = 1 + S(A|B)), and the tightness of the inequality (δ = U - B). Numerical evaluations reveal a nuanced and non-universal dependence on acceleration. Contrary to the simple intuition that acceleration (Unruh temperature) should always degrade correlations and thus worsen (increase) the uncertainty bound, the results show that the bound’s behavior critically depends on the initial state (Δ0).

For an initial state with Δ0 = -1, both U and B increase with T, but their difference δ decreases, meaning the uncertainty relation becomes tighter. For Δ0 = 0.5, U increases monotonically, but B exhibits a non-monotonic response, causing the tightness δ to first approach zero (saturation) and then increase again. For a maximally correlated initial state (Δ0 = 1), the relation is perfectly tight (δ=0) at zero temperature but becomes progressively looser (δ increases) as T rises, even though the uncertainty U grows.

To unravel this complexity, the authors decompose the lower bound into information-theoretic components: B = log₂(1/c) + M - D, where M is the “minimum missing information” about the measured system after an optimal measurement on the memory, and D is the quantum discord (a measure of non-classical correlations). By examining the evolution of M and D with T for different Δ0 values, they demonstrate that the behavior of the uncertainty bound is governed by the competition between these two quantities. For instance, in some regimes, an increase in quantum discord (D) is accompanied by a larger increase in missing information (M), leading to a net increase in the bound B and uncertainty U. This key finding challenges the straightforward notion that stronger quantum correlations invariably lead to lower measurement uncertainty, highlighting a subtle interplay unique to relativistic quantum open systems.

In conclusion, this study provides a comprehensive framework for analyzing quantum uncertainty in accelerated frames. It establishes that the QMA-EUR is not merely degraded by relativistic motion but exhibits rich, state-dependent dynamics. The work suggests that entropic uncertainty relations can serve as sensitive probes for studying quantum information flow and correlation degradation in extreme spacetimes, with potential implications for understanding physics near black hole horizons or in early universe cosmology.