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Communications in Theoretical Physics

Complementary relation between quantum entanglement and entropic uncertainty

Yun Cao, Dong Wang, Xiao‐Gang Fan, Fei Ming, Zhang-Yin Wang, Ye Liu

Published Dec 18, 2020• 11 citations
Publisher

Abstract

Abstract Quantum entanglement is regarded as one of the core concepts, which is used to describe the non-classical correlation between subsystems, and entropic uncertainty relation plays a vital role in quantum precision measurement. It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states. An interesting question is naturally raised: is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement? Or if the relation can be applied to estimate the entanglement. In this work, we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation. The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system, and specifically the larger uncertainty will induce the weaker entanglement of the probed system, and vice versa. Besides, we use randomly generated states as illustrations to verify our results. Therefore, we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement.

DOI
10.1088/1572-9494/abc46f

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Abstract Quantum entanglement is regarded as one of the core concepts, which is used to describe the non-classical correlation between subsystems, and entropic uncertainty relation plays a vital role in quantum precision measurement. It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states. An interesting question is naturally raised: is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement? Or if the relation can be applied to estimate the entanglement. In this work, we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation. The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system, and specifically the larger uncertainty will induce the weaker entanglement of the probed system, and vice versa. Besides, we use randomly generated states as illustrations to verify our results. Therefore, we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement.
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.
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.
The entropic uncertainty principle in the form proven by Maassen and Uffink yields a fundamental inequality that is prominently used in many places all over the field of quantum information theory. In this paper, we provide a family of versatile generalizations of this relation. Our proof methods build on a deep connection between entropic uncertainties and interpolation inequalities for the doubly stochastic map that links probability distributions in two measurement bases. In contrast to the original relation, our generalization also incorporates the von Neumann entropy of the underlying quantum state. These results can be directly used to bound the extractable randomness of a source-independent quantum random number generator in the presence of fully quantum attacks, to certify entanglement between trusted parties, or to bound the entanglement of a system with an untrusted environment.
The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find entanglement-dependent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem 1 . Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on the amount of extractable key in a quantum cryptographic scenario.
The Heisenberg uncertainty principle is one of the characteristics of quantum mechanics. With the vigorous development of quantum information theory, uncertain relations have gradually played an important role in it. In particular, in order to solved the shortcomings of the concept in the initial formulation of the uncertainty principle, we brought entropy into the uncertainty relation, after that, the entropic uncertainty relation has exploited the advantages to the full in various applications. As we all know the entropic uncertainty relation has became the core element of the security analysis of almost all quantum cryptographic protocols. This review mainly introduces development history and latest progress of uncertain relations. After Heisenberg's argument that incompatible measurement results are impossible to predict, many scholars, inspired by this viewpoint, have made further relevant investigations. They combined the quantum correlation between the observable object and its environment, and carried out various generalizations of the uncertainty relation to obtain more general formulas. In addition, it also focuses on the entropy uncertainty relationship and quantum-memory-assisted entropic uncertainty relation, and the dynamic characteristics of uncertainty in some physical systems. Finally, various applications of the entropy uncertainty relationship in the field of quantum information are discussed, from randomnesss to wave-particle duality to quantum key distribution.
The uncertainty principle is one of the comprehensive and fundamental concept in quantum theory. This principle states that it is not possible to simultaneously measure two incompatible observatories with high accuracy. Uncertainty principle has been formulated in various form. The most famous type of uncertainty relation is expressed based on the standard deviation of observables. In quantum information theory the uncertainty principle can be formulated using Shannon and von Neumann entropy. Entropic uncertainty relation in the presence of quantum memory is one of the most useful entropic uncertainty relations. Due to their importance and scalability, solid state systems have received considerable attention nowadays. In this work we will consider a quantum dot system as a solid state system. We will study the quantum correlation and quantum memory assisted entropic uncertainty in this typ of system. We will show that the temperature in of quantum dot system can affect the quantum correlation and entropic uncertainty bound. It will be observed that the entropic uncertainty bound decreases with decreasing temperature and quantum correlations decreases with increasing the temperature.