Abstract We derive a generalized uncertainty relation (GUR) in the entropic framework for multipartite identical pure entangled systems. Starting from the variance-based GUR, we reformulate it in terms of differential Shannon entropy, obtaining a tighter entropic bound that explicitly accounts for quantum correlations. Our analysis reveals that the conditional entropy , which quantifies the entanglement between a measured subsystem k and the remaining particles acting as a composite quantum memory, decreases with particle number N and asymptotically approaches zero. This behavior highlights the entanglement-induced suppression of uncertainty, leading to an emergent classicality in the large- N limit. In particular, we demonstrate that multipartite entanglement with quantum memory reduces both the state-independent and state-dependent bounds, yielding more precise measurements. These results not only provide a deeper understanding of the interplay between entanglement and uncertainty but also suggest potential applications in quantum cryptography, precision metrology, condensed-matter many-body physics, quantum thermodynamics, and foundational studies of the quantum-to-classical transition.
Abstract We derive a generalized uncertainty relation (GUR) in the entropic framework for multipartite identical pure entangled systems. Starting from the variance-based GUR, we reformulate it in terms of differential Shannon entropy, obtaining a tighter entropic bound that explicitly accounts for quantum correlations. Our analysis reveals that the conditional entropy , which quantifies the entanglement between a measured subsystem k and the remaining particles acting as a composite quantum memory, decreases with particle number N and asymptotically approaches zero. This behavior highlights the entanglement-induced suppression of uncertainty, leading to an emergent classicality in the large- N limit. In particular, we demonstrate that multipartite entanglement with quantum memory reduces both the state-independent and state-dependent bounds, yielding more precise measurements. These results not only provide a deeper understanding of the interplay between entanglement and uncertainty but also suggest potential applications in quantum cryptography, precision metrology, condensed-matter many-body physics, quantum thermodynamics, and foundational studies of the quantum-to-classical transition.
Abstract Entropic uncertainty relation (EUR) plays a vital role in quantum information theories by demonstrating the intrinsic uncertainty of nature from the information-theoretic perspective. A tighter lower bound for uncertainty relations facilitates more accurate predictions of measurement outcomes and more robust quantum information processing. However, meaningful EURs in multipartite scenarios are yet to be identified. In this study, we derived a generalized EUR (GEUR) for the measurement of multiple observables in arbitrary many-body systems. The corresponding GEUR bound is shown to be tighter than Renes--Boileau \textit{et al.}'s bound when applied in a tripartite scenario. We experimentally tested the GEUR with a four-photon entangled state in an optical platform. We demonstrated that GEUR provides a tighter lower bound of the quantum-secret-key rate, which can readily accelerate practical quantum key distribution protocols. By providing a close peek at the nature of uncertainty, our results may find broad applications in the security analysis of quantum cryptography.
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
The uncertainty principle, which demonstrates the intrinsic uncertainty of nature from an information-theory perspective, is at the heart of quantum information theory. In the realm of quantum information theory, Shannon entropy is used to depict the uncertainty relation in general. A tighter lower bound for uncertainty relations facilitates more accurate predictions of measurement outcomes and more robust quantum information processing. Interestingly, the tripartite entropic uncertainty relation (EUR) can be further optimized. Renes et al. proposed a tripartite EUR [J. M. Renes and J.-C. Boileau, Phys. Rev. Lett. 103, 020402 (2009)], and subsequently, Ming et al. strengthened its lower bound in [F. Ming, D. Wang, X.-G. Fan, W.-N. Shi, L. Ye, and J.-L. Chen, Phys. Rev. A 102, 012206 (2020)]. Specifically, we derive a tighter lower bound of the tripartite EUR using the Holevo quantity. Furthermore, we generalize the tripartite EUR, that is, the generalized entropic uncertainty relation for multiple measurements in multipartite systems. As illustrations, we provide several typical examples to show that our bound is tight and outperforms the previous bound. Furthermore, our findings pave the way for using the tighter bound for the quantum secret key rate in quantum key distribution protocols and are essential for quantum precision measurements in the framework of genuine multipartite systems. By providing a close peek at the nature of uncertainty, our results may find broad applications in the security analysis of quantum cryptography.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
Entropic uncertainty relations demonstrate the intrinsic uncertainty of nature from an information-theory perspective. Recently, a quantum-memory-assisted entropic uncertainty relation for multiple measurements was proposed by Wu et al. [Phys. Rev. A 106, 062219 (2022)]. Interestingly, the quantum-memory-assisted entropic uncertainty relation for multiple measurement settings can be further generalized. In this work, we propose two complementary multipartite quantum-memory-assisted entropic uncertainty relations and our lower bounds depend on values of complementarity of the observables, (conditional) von Neumann entropies, Holevo quantities, and mutual information. As an illustration, we provide several typical cases to exhibit that our bounds are tighter and outperform the previous bounds.
The uncertainty principle is a fundamental principle in quantum physics. It implies that the measurement outcomes of two incompatible observables cannot be predicted simultaneously. In quantum information theory, this principle can be expressed in terms of entropic measures. M. Berta et al. [Nat. Phys. 6, 659 (2010)] have indicated that uncertainty bound can be altered by considering a particle as a quantum memory correlating with the primary particle. In this article, we obtain a lower bound for entropic uncertainty in the presence of a quantum memory by adding an additional term depending on the Holevo quantity and mutual information. We conclude that our lower bound will be tightened with respect to that of Berta et al. when the accessible information about measurements outcomes is less than the mutual information about the joint state. Some examples have been investigated for which our lower bound is tighter than Berta et al.'s lower bound. Using our lower bound, a lower bound for the entanglement of formation of bipartite quantum states has been obtained, as well as an upper bound for the regularized distillable common randomness.