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Scientific Reports

Coherent states and unilateral coherence in strengthening entropic uncertainty relations

Forough Panahyazdan, Ahmad Akhound

Published Apr 12, 2025• 2 citations
Publisher

Abstract

In this paper, we examine entropy uncertainty relations in the presence of quantum memory and quantum entanglement for non-orthogonal states constructed from two-qubit coherent states. Our results reveal that for the studied state, the entropy uncertainty relations under quantum memory conditions remain tight across a broad range of parameters, leading to lower error rates and higher precision in observable predictions. We explicitly demonstrate that in the presence of quantum memory, the ability to predict observables is fundamentally tied to entanglement. Stronger entanglement leads to reduced entropy uncertainty and greater predictive accuracy. On the other hand, as entanglement weakens, even though the uncertainty relations remain tight, prediction errors increase accordingly. In the maximum entanglement state, the accuracy of the predictions is also maximized. As the entanglement decreases, despite the tightening of the uncertainty relations, the accuracy of the predictions decreases. The prediction accuracy is fully proportional to the amount of entanglement, and it is possible to achieve the desired minimum entropy uncertainty and high correlation according to the problem's requirements. This is while the numerical values of the components forming the upper bound of the entropy uncertainty relations are not equal in all cases. With respect to the changes examined from the presented state, quantum correlation, entropy uncertainty, and the components forming the upper bound, regardless of the amount of entanglement, each separately tend to a constant value.

DOI
10.1038/s41598-025-95598-3

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In this paper, we examine entropy uncertainty relations in the presence of quantum memory and quantum entanglement for non-orthogonal states constructed from two-qubit coherent states. Our results reveal that for the studied state, the entropy uncertainty relations under quantum memory conditions remain tight across a broad range of parameters, leading to lower error rates and higher precision in observable predictions. We explicitly demonstrate that in the presence of quantum memory, the ability to predict observables is fundamentally tied to entanglement. Stronger entanglement leads to reduced entropy uncertainty and greater predictive accuracy. On the other hand, as entanglement weakens, even though the uncertainty relations remain tight, prediction errors increase accordingly. In the maximum entanglement state, the accuracy of the predictions is also maximized. As the entanglement decreases, despite the tightening of the uncertainty relations, the accuracy of the predictions decreases. The prediction accuracy is fully proportional to the amount of entanglement, and it is possible to achieve the desired minimum entropy uncertainty and high correlation according to the problem's requirements. This is while the numerical values of the components forming the upper bound of the entropy uncertainty relations are not equal in all cases. With respect to the changes examined from the presented state, quantum correlation, entropy uncertainty, and the components forming the upper bound, regardless of the amount of entanglement, each separately tend to a constant value.
The uncertainty principle sets a bound on our ability to predict the measurement outcomes of two incompatible observables which are measured on a quantum particle simultaneously. In quantum information theory, the uncertainty principle can be formulated in terms of the Shannon entropy. Entropic uncertainty bound can be improved by adding a particle which correlates with the measured particle. The added particle acts as a quantum memory. In this work, a method is provided for obtaining the entropic uncertainty relations in the presence of a quantum memory by using quantum coherence. In the method, firstly, one can use the quantum relative entropy of quantum coherence to obtain the uncertainty relations. Secondly, these relations are applied to obtain the entropic uncertainty relations in the presence of a quantum memory. In comparison with other methods this approach is much simpler. Also, for a given state, the upper bounds on the sum of the relative entropies of unilateral coherences are provided, and it is shown which one is tighter. In addition, using the upper bound obtained for unilateral coherence, the nontrivial upper bound on the sum of the entropies for different observables is derived in the presence of a quantum memory.
Abstract In this paper, a method is provided for obtaining the entropic uncertainty relations in the presence of a quantum memory by using quantum coherence. In the method, firstly, one can use the quantum relative entropy of quantum coherence to obtain the uncertainty relations. Secondly, these relations are applied to obtain the entropic uncertainty relations in the presence of a quantum memory. In comparison with other methods this approach is much simpler. Also, for a given state, the upper bounds on the sum of the relative entropies of unilateral coherences are provided, and it is shown which one is tighter. In addition, using the upper bound obtained for unilateral coherence, the nontrivial upper bound on the sum of the entropies for different observables is derived in the presence of a quantum memory.
We investigate the dynamics of entropic uncertainty relations, tightness, and concurrence in two non-interacting qubits produced in the Werner state and subjected to classical channels. In two contexts, common and independent qubit-noise configurations, a Gaussian Ornstein Uhlenbeck process is used to control the noisy effects of the local external fields. Using parameter optimization, we establish long-term stability in two qubits while reducing the environment's disorder entropic impact and maintaining entanglement. Using entropic uncertainty relations, tightness and concurrence, we show that in the presence of Ornstein Uhlenbeck noise, two-qubit coherence and entanglement in local fields are both weak and readily lost. Despite this, longer entanglement and lower entropy can be predicted within a limited range of the noise parameter and the purity estimator of the two-qubit state. Furthermore, the measured rate of entropy rise outpaces the rate of disentanglement generation. In addition, we also utilized the tightness measure to estimate the uncertainty in the dynamics of bipartite entropic relations.
A method is presented by which uncertainty relations in the absence of quantum memory can be converted to those in the presence of quantum memory. It is shown that the lower bounds obtained through the method are tighter than those having been achieved so far. The method is also used to obtain the uncertainty relations for multiple measurements in the presence of quantum memory. For a given state, the lower bounds on the sum of the relative entropies of unilateral coherence are provided using uncertainty relations in the presence of quantum memory, and it is shown which bound is tighter.
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.
The uncertainty principle limits the ability for observer to precisely measure two incompatible observables, and plays a crucial role in quantum precision measurement in the quantum information science. When quantum systems interact with their surroundings, they inevitably result in decoherence, which increases the uncertainty of the system. In the process of quantum information processing, the effective regulation of uncertainty becomes a key problem that needs to be solved. In this work, we investigate the quantum-memory-assisted entropic uncertainty relation of a two-qubit system under correlated channels with dephasing colored noise. We demonstrate that it is possible to control the entropic uncertainty, U, and its lower bound, Ub, by combining correlations between successive uses of channels and the non-Markovianity of the dynamical evolution. Firstly, the evolutionary characteristics of the trace distance are employed to distinguish between Markovianity and non-Markovianity of the channel. Subsequently, the system is selected to be either a maximally entangled state or separated state initially. By adjusting the strength η of the correlations, we find that with the increase of η, the entropic uncertainty and its lower bound decrease. Especially, if the channel is fully correlated (η = 1), the entropic uncertainty and its lower bound remain constant under the channel, indicating that decoherence is completely suppressed. A comparison of Markovian channel with non-Markovian channel reveals that the entropic uncertainty and its lower bound exhibit oscillatory behaviour under non-Markovian channels. The combination of correlations and non-Markovianity of the channel demonstrates that the entropic uncertainty and its lower bound can be reduced under fully correlated channels where the non-Markovianity has no effect. This is because fully correlated channels suppress decoherence to a greatest extent. Under partially correlated channels, the combination of correlations and non-Markovianity can more effectively reduce the entropic uncertainty and its lower bound. Under such channels, correlations of the channel reduce the entropic uncertainty and its lower bound during the whole evolution, while the non-Markovianity contributes to their oscillations and reduce them in some specific time. Furthermore, the results show that the entropic uncertainty and its lower bound reach steady values that depend only on the strength of the correlations after long-time evolution. In other words, the stronger the correlations, the lower the entropy uncertainty and its lower bound of steady states will be. Finally, we analyse the physical nature of the decrease of the entropic uncertainty and its lower bound, and it is found that the decrease of the entropic uncertainty and its lower bound originate from the increase of the quantum correlations in the system.