Generalized entropic uncertainty relation and non-classicality in Schwarzschild black hole

TL;DR

The paper introduces a generalized quantum-memory-assisted entropic uncertainty relation (QMA-EUR) for arbitrary multi-observable measurements in multipartite systems by leveraging the Holevo quantity, and demonstrates a tighter, scalable bound (Eq. 207) that outperforms previous formulations. It then applies this framework to Schwarzschild spacetime, analyzing Dirac vacuum structures, Hawking-Unruh effects, and information redistribution between accessible and inaccessible regions. A key finding is the exact equivalence between $N$-partite GM concurrence and the $l_1$-norm coherence for GHZ-type states in this setting, and the observation that quantum coherence diminishes while entropy-based uncertainty grows with increasing Hawking temperature. The study examines GHZ-type and Werner initial states to show the generality of the tighter EUR bound in curved spacetime and reveals an anti-correlation between EUR and coherence, highlighting deep connections between quantum resources and relativistic gravity phenomena.

Abstract

The uncertainty principle constitutes a fundamental pillar of quantum theory, representing one of the most distinctive features that differentiates quantum mechanics from classical physics. In this study, we firstly propose a novel generalized entropic uncertainty relation (EUR) for arbitrary multi-measurement in the many-body systems, and rigorously derive a significantly tighter bound compared to existing formulations. Specifically, we discuss the proposed EUR in the context of Schwarzschild black hole, where we demonstrate the superior tightness of our derived bound. The study further elucidates the dynamical evolution of multipartite quantum coherence and entanglement in the curved spacetime. A particularly noteworthy finding reveals the exact equivalence between entanglement and $l_1$-norm coherence for arbitrary $N$-partite Greenberger-Horne-Zeilinger-type (GHZ-type) states. Moreover, we find that quantum coherence is significantly diminished and the measurement uncertainty increases to a stable maximum with increasing Hawking temperature. Thus, the findings of this study contribute to a deeper understanding of non-classicality and quantum resources in black holes.

Figures (6)

• Figure 1: The Penrose diagram shows the trajectories for Bob and Anti-Bob. ${i^0}$ denotes the spatial infinities, ${i^-}$ and ${i^+}$ represents the timelike past and future infinities respectively; ${j^-}$ and ${j^+}$ respectively represents the null past and future infinities and ${H^ \pm }$ amount to $r_{0} = 2M$ and denote the event horizons
• Figure 2: $l_1$-norm ${C_{{l_1}}}({\hat{\rho} _{N - n,p,q}})$ as functions of $T$ and $R_0$ for different $p$, $q$ and $n$. Graphs (a) and (d) correspond to the case where $p = n$ and $q = 0$. Graphs (b) and (e) represent the case where $p > q$ and graphs (c) and (f) represent the case of $p < n$ respectively.
• Figure 3: Graph (a) shows the uncertainty and two bounds with distance $R_0$ for $\Omega = \omega_k / T = 30$ and $z = \frac{\sqrt{2}}{2}$. Graph (b) describes the uncertainty and two bounds with Hawking Temperature $T$ with ${\omega _k} = 1$, ${{R_0} = r \mathord{\left/ {\newline} \right. \nulldelimiterspace} {{R_H} = 1.05}}$ and $z = \frac{{\sqrt 2 }}{2}$. Here, $U$ represents the entropic uncertainty, Bound1 is the lower bound proposed by Renes and Boileau JMR, and Bound2 is our proposed new lower bound. Graphs (c) and (d) plot the $l_1$-norm as functions of $R_0$ and $T$ respectively.
• Figure 4: Entropic uncertainty and coherence with respect to distance $r$ and Hawking Temperature $T$ of Werner state $\hat{\rho} _{A{B\rm{_I}}{C\rm{_I}}}^{\rm Werner}$ in Schwarzschild black hole. The system purity $p = 0.5$ and state parameter $z = \frac{{\sqrt 2 }}{2}$. Graph (a) and Graph (b) describe the relationship of EURs with distance and temperature respectively. In these graphs, U, Bound1, and Bound2 denote the entropic uncertainty, the lower bound proposed by Renes, and our newly proposed lower bound respectively. Graph (a) presents the EUR vs distance $R_0$ for $\Omega = \omega_k / T = 30$ and $z = \frac{\sqrt{2}}{2}$. Graph (b) describes the EUR with Hawking Temperature $T$ with ${\omega _k} = 1$, ${{R_0} = r \mathord{\left/ {\newline} \right. \nulldelimiterspace} {{R_H} = 1.05}}$ and $z = \frac{{\sqrt 2 }}{2}$. Graph (c) and (d) present the $l_1$-norm as the function of $R_0$ and $T$ respectively.
• Figure 5: The bound gain ($\Delta={\rm Bound}2-{\rm Bound}1$) when the initial states are chosen to GHZ-type and Werner states in the Schwarzschild black hole. Graphs (a), (b) and (c) show the bound gain versus $R_0$, $T$ and $z$ of GHZ-type states, while Graphs (d), (e) and (f) represent the the Werner states respectively. Graphs (a) and (d): $\Delta$ vs $R_0$ with $\Omega = 30$ and ${z^2} = \frac{1}{2}$; Graphs (b) and (e): $\Delta$ vs $T$ with $R_0 = 1.05$ and ${z^2} = \frac{1}{2}$; Graphs (c) and (f): $\Delta$ vs $z$ with $\Omega = 1$ and $R_0 = 1.05$.