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arXiv:2602.11503 · Physical review. D/Physical review. D.

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

Rui-Jie Yao, Dong Wang

gr-qchep-thquant-phSubmitted Feb 12, 2026• Announced Feb 13, 2026• 2 citations
arXivHTMLPDF

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.

Journal reference
Phys. Rev. D 113, 065002 (2026)
DOI
10.1103/ydtw-kwzf
Comments
18 pages, 6 figures. Comments are welcomed. Accepted by Physical Review D

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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 l1l_1l1​-norm coherence for arbitrary NNN-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.
The uncertainty principle is deemed as one of cornerstones in quantum mechanics, and exploring its lower limit of uncertainty will be helpful to understand the principle's nature. In this study, we propose a generalized entropic uncertainty relation for arbitrary multiple-observable in multipartite system, and further derive a tighter lower bound by considering Holevo quality and mutual information. Importantly, we detailedly discuss the proposed uncertainty relations and quantum coherence in the context of Schwarzschild space-time. It is interesting to find that Hawking radiation will damage the coherence of the physically accessible region and increase the uncertainty. Furthermore, we argue that the properties of the uncertainty in Schwarzschild space-time can be explained from the systems' purity and the information redistribution of the different regions. Therefore, it is believed that our findings provide the generalized entropic uncertainty relations in multipartite systems, which may facilitate us deeper understanding of quantumness and information paradox of the black holes.
We explore the entropic uncertainty relation in the curved background outside a Schwarzschild black hole , and find that Hawking radiation introduces a nontrivial modification on the uncertainty bound for particular observer, therefore it could be witnessed by proper uncertainty game experimentally. We first investigate an uncertainty game between a free falling observer and his static partner holding a quantum memory initially entangled with the quantum system to be measured. Due to the information loss from Hawking decoherence, we find an inevitable increase of the uncertainty on the outcome of measurements in the view of static observer, which is dependent on the mass of the black hole, the distance of observer from event horizon , and the mode frequency of quantum memory. To illustrate the generality of this paradigm, we relate the entropic uncertainty bound with other uncertainty probe, e.g., time–energy uncertainty. In an alternative game between two static players, we show that quantum information of qubit can be transferred to quantum memory through a bath of fluctuating quantum fields outside the black hole. For a particular choice of initial state, we show that the Hawking decoherence cannot counteract entanglement generation after the dynamical evolution of system, which triggers an effectively reduced uncertainty bound that violates the intrinsic limit − log 2 ⁡ c . Numerically estimation for a proper choice of initial state shows that our result is comparable with possible real experiments. Finally, a discussion on the black hole firewall paradox in the context of entropic uncertainty relation is given.
For a pair of incompatible quantum measurements, the total uncertainty can be bounded by a state-independent constant. However, such a bound can be violated if the quantum system is entangled with another quantum system (called memory); the quantum correlation between the systems can reduce the measurement uncertainty. On the other hand, in a curved spacetime, the presence of the Hawking radiation can reduce quantum correlation. The interplay of quantum correlation in the curved spacetime has become an interesting arena for studying uncertainty relations. Here we demonstrate that the bounds of the entropic uncertainty relations, in the presence of memory, can be formulated in terms of the Holevo quantity, which limits how much information can be encoded in a quantum system. Specifically, we considered examples with Dirac fields with and without spin, near the event horizon of a Schwarzschild black hole, the Holevo bound provides a better bound than the previous bound based on the mutual information. Furthermore, as the memory moves away from the black hole, the difference between the total uncertainty and the new lower bound remains a constant, not depending on any property of the black hole.
Abstract The Heisenberg uncertainty principle describes a basic restriction on an observer's ability of precisely predicting the measurement of a pair of noncommuting observables, and virtually is at the core of quantum mechanics. Herein, the aim is to study the entropic uncertainty relation (EUR) under the background of a Schwarzschild black hole and its control. Explicitly, dynamical features of the measuring uncertainty via entropy are developed in a practical model where a stationary particle interacts with its surrounding environment while another particle—serving as a quantum memory reservoir—undergoes free fall in the vicinity of the event horizon of the Schwarzschild space‐time. It shows higher Hawking temperatures would give rise to an inflation of the entropic uncertainty on the measured particle. This is suggestive of the fact the measurement uncertainty is strongly correlated with degree of mixing present in the evolving particles. Additionally, based on information flow theory, a physical interpretation for the observed dynamical behaviors related with the entropic uncertainty in such a genuine scenario is provided. Finally, an efficient strategy is proposed to reduce the uncertainty by non‐tracing‐preserved operations. Therefore, our explorations may improve the understanding of the dynamic entropic uncertainty in a curved space‐time, and illustrate predictions of quantum measurements in relativistic quantum information sciences.
Heisenberg's uncertainty principle is a fundamental element in quantum mechanics. It sets a bound on our ability to predict the measurement outcomes of two incompatible observables simultaneously. In quantum information theory, the uncertainty principle can be expressed using entropic measures. The entropic uncertainty relation can be improved by considering an additional particle as a memory particle. The presence of quantum correlation between the memory particle and the measured particle reduces the uncertainty. In a curved space-time, the presence of the Hawking radiation can reduce quantum correlation. Therefore, concerning the relationship between the quantum correlation and entropic uncertainty lower bound, we expect that the Hawking radiation increases the entropic uncertainty lower bound. In this work, we investigate the entropic uncertainty relation in Garfinkle-Horowitz-Strominger (GHS) dilation black hole. We consider a model in which the memory particle is located near the event horizon outside the black hole, while the measured particle is free falling. To study the proposed model, we will consider examples with Dirac fields. We also explore the effect of the Hawking radiation on the quantum secret key rate.
Abstract The uncertainty relation is regarded as a remarkable feature of quantum mechanics differing from the classical counterpart, and it plays a backbone role in the region of quantum information theory. In principle, the uncertainty relation offers a nontrivial limit to predict the outcome of arbitrarily incompatible observed variables. Therefore, to pursue a more general uncertainty relations ought to be considerably important for obtaining accurate predictions of multi-observable measurement results in genuine multipartite systems. In this article, we derive a generalized entropic uncertainty relation (EUR) for multi-measurement in a multipartite framework. It is proved that the bound we proposed is stronger than the one derived from Renes et al. in [Phys. Rev. Lett. 103,020402(2009) ] for the arbitrary multipartite case. As an illustration, we take several typical scenarios that confirm that our proposed bound outperforms that presented by Renes et al. Hence, we believe our findings provide generalized uncertainty relations with regard to multi-measurement setting, and facilitate the EUR’s applications on quantum precision measurement regarding genuine multipartite systems.