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

TL;DR

The paper studies how uniform acceleration and the resulting Unruh effect impact the quantum-memory-assisted entropic uncertainty relation (QMA-EUR) for two Unruh–DeWitt detectors in $(3+1)$-D Minkowski space. Using a Markovian Kossakowski–Lindblad framework, it derives a stationary, X-shaped two-qubit state whose parameters depend on the Unruh temperature $T$, the detector gap $\omega$, and the initial correlation parameter $\Delta_0$, and computes the entropic uncertainties $U$, the bound $\mathcal{B}$, and the tightness $\delta$. The results show that acceleration can both tighten and loosen the EUR bound depending on $\Delta_0$, with the behavior explained by the interplay between quantum discord $D$ and the minimum missing information $M$ via $\mathcal{B} = \log_2(1/c) + M - D$, where $c = 1/2$ for mutually unbiased measurements. Importantly, the study reveals that higher quantum discord does not necessarily reduce uncertainty, highlighting nuanced relativistic effects on quantum information measures and informing relativistic quantum information protocols.

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.

Figures (2)

• Figure 1: The uncertainty $U$, its bound $\mathcal{B}$ and its tightness $\delta$ as a function of the Unruh temperature $T$, for different values of the initial state selection parameter: (a)$\Delta_0 =-1$; (b)$\Delta_0 =0.5$; (c)$\Delta_0 =1$. In all numerics we set $\omega=1$.
• Figure 2: The dynamics of the quantum discord and the minimal missing information as a function of the Unruh temperature $T$, for different values of the initial state parameter: (a)$\Delta_0 =-1$; (b)$\Delta_0 =0.5$; (c)$\Delta_0 =1$. In all numerics we set $\omega=1$.