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arXiv:2004.09736 · Physical review. D/Physical review. D.

Multiple uncertainty relation for accelerated quantum information

Chen Qian, Yadong Wu, Jia-Wei Ji, Yunlong Xiao, Barry C. Sanders

Published Nov 10, 2020• 6 citations
arXivHTMLPDF

Abstract

The uncertainty principle, first introduced by Heisenberg in inertial frames, clearly distinguishes quantum theories from classical mechanics. In noninertial frames, its information-theoretic expressions, namely, entropic uncertainty relations, have been extensively studied through delocalized quantum fields, and localization of the quantum fields were discussed as well. However, infeasibility of measurements applied on a delocalized quantum field due to the finite size of measurement apparatuses is left unexplained. Therefore, physical clarification of a quantum protocol revealing entropic uncertainty relations still needs investigation. Building on advances in quantum field theories and theoretical developments in entropic uncertainty relations, we demonstrate a relativistic protocol of an uncertainty game in the presence of localized fermionic quantum fields inside cavities. Moreover, a novel lower bound for entropic uncertainty relations with multiple quantum memories is given in terms of the Holevo quantity, which implies how acceleration affects uncertainty relations.

DOI
10.1103/physrevd.102.096009

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The uncertainty principle, first introduced by Heisenberg in inertial frames, clearly distinguishes quantum theories from classical mechanics. In noninertial frames, its information-theoretic expressions, namely, entropic uncertainty relations, have been extensively studied through delocalized quantum fields, and localization of the quantum fields were discussed as well. However, infeasibility of measurements applied on a delocalized quantum field due to the finite size of measurement apparatuses is left unexplained. Therefore, physical clarification of a quantum protocol revealing entropic uncertainty relations still needs investigation. Building on advances in quantum field theories and theoretical developments in entropic uncertainty relations, we demonstrate a relativistic protocol of an uncertainty game in the presence of localized fermionic quantum fields inside cavities. Moreover, a novel lower bound for entropic uncertainty relations with multiple quantum memories is given in terms of the Holevo quantity, which implies how acceleration affects uncertainty relations.
Abstract The principles of quantum information provide new avenues to investigate the cosmos. The uncertainty principle is an important trait of the nonclassical world, and it characterizes a significant lower bound (LB) that can be used to estimate measurement results for two noncommuting observables. Subsequently, the uncertainty principle is generalized to a new version by using the entropy and the quantum memory, that is, the quantum-memory-assisted entropic uncertainty relations (Q-M-A-E-U-Rs). Here, considering two qubit detectors coupled to scalar fields, we explore the effects of different cosmic parameters on Q-M-A-E-U-Rs and reveal the influence of the Holevo quantity on the LB of a Q-M-A-E-U-R. It is revealed that an increase in the expansion rapidity of spacetime can enhance the entropic uncertainty and decrease the ability to accurately predict the measurement outcome. The volume expansion leads to the invariance of the entropic uncertainty. An increase in the particle mass of the scalar field causes degradation in entropic uncertainty. In addition, the influence of the Holevo quantity on the Q-M-A-E-U-R’s LB can be ignored if one considers (σx,σy)({\sigma _x},{\sigma _y})(σx​,σy​) (σx,σy) as two complementary observables. Therefore, one can use the Adabi bound and Berta bound to equivalently predict the left-hand side (LHS) of the Q-M-A-E-U-R in this situation. However, when (σx,σz)({\sigma _x},{\sigma _z})(σx​,σz​) (σx,σz) are chosen as two noncommuting observables, the effect of the Holevo quantity on the LB of the uncertainty relation cannot be ignored, and the Adabi bound can always precisely achieve the LHS and predict entropic uncertainty.
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.
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.
One unique feature of quantum mechanics is the Heisenberg uncertainty principle, which states that the outcomes of two incompatible measurements cannot simultaneously achieve arbitrary precision. In an information-theoretic context of quantum information, the uncertainty principle can be formulated as entropic uncertainty relations with two measurements for a quantum bit (qubit) in two-dimensional system. New entropic uncertainty relations are studied for a higher-dimensional quantum state with multiple measurements, and the uncertainty bounds can be tighter than that expected from two measurements settings and cannot result from qubits system with or without a quantum memory. Here we report the first room-temperature experimental testing of the entropic uncertainty relations with three measurements in a natural three-dimensional solid-state system: the nitrogen-vacancy center in pure diamond. The experimental results confirm the entropic uncertainty relations for multiple measurements. Our result represents a more precise demonstrating of the fundamental uncertainty principle of quantum mechanics.
The uncertainty principle is a striking and fundamental feature in quantum mechanics, distinguishing it from classical mechanics. It offers an important lower bound to predict outcomes of two arbitrary incompatible observables measured on a particle. In quantum information theory, this uncertainty principle is popularly formulized in terms of entropy. Here, we present an improvement of the tripartite quantum-memory-assisted entropic uncertainty relation. The uncertainty's lower bound is derived by considering mutual information and the Holevo quantity. It shows that the bound derived by this method will be tighter than the lower bound of Renes and Boileau [Phys. Rev. Lett. 103, 020402 (2009)]. Furthermore, regarding a pair of mutual unbiased bases as the incompatibility, our bound will become extremely tight for the three-qubit XXX-state system, completely coinciding with the entropy-based uncertainty, and can restore Renes and Boileau's bound with respect to arbitrary tripartite pure states. In addition, by applying our lower bound one can attain the tighter bound of the quantum secret key rate, which is of basic importance to enhance the security of quantum key distribution protocols.
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.