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Annalen der Physik

Dynamics of Quantum Correlation and Entropic Uncertainty in Spin‐12$\frac{1}{2}$ Alternating Transverse Ising Model

Tian‐Yi Jiang, Yu‐Yan Fang, Yinghong Li, Xinye Xu, Jin‐Ming Liu

Published Dec 16, 2021• 9 citations
Publisher

Abstract

Abstract By implementing the quantum renormalization group method, the effective Hamiltonian of the spin‐ Ising model with regularly altering strengths of both the magnetic fields and the exchange couplings are first derived analytically. Then, tripartite quantum discord (QD) and entropic uncertainty (EU) of the Ising model and their quantum phase transitions are investigated. Moreover, the quantum critical points and correlation length exponents of the model are analyzed, and both the block–block EU and the site–site EU are discussed through numerical calculations. It is found that under the system size increasing, the first derivatives of both tripartite QD and block–block EU show nonanalytic and scaling behaviors near the critical point. Furthermore, the strong quantum correlations among multi‐particles do not necessarily lead to the decrease of EU, as the site–site EU in the vicinity of the critical point increases due to competitions among quantum correlations. The findings may offer new insights into utilizing EU as an observable of the quantum phase transitions.

DOI
10.1002/andp.202100352

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Dynamics of Quantum Correlation and Entropic Uncertainty in Spin‐1212\frac{1}{2}21​ Alternating Transverse Ising Model
Abstract By implementing the quantum renormalization group method, the effective Hamiltonian of the spin‐ Ising model with regularly altering strengths of both the magnetic fields and the exchange couplings are first derived analytically. Then, tripartite quantum discord (QD) and entropic uncertainty (EU) of the Ising model and their quantum phase transitions are investigated. Moreover, the quantum critical points and correlation length exponents of the model are analyzed, and both the block–block EU and the site–site EU are discussed through numerical calculations. It is found that under the system size increasing, the first derivatives of both tripartite QD and block–block EU show nonanalytic and scaling behaviors near the critical point. Furthermore, the strong quantum correlations among multi‐particles do not necessarily lead to the decrease of EU, as the site–site EU in the vicinity of the critical point increases due to competitions among quantum correlations. The findings may offer new insights into utilizing EU as an observable of the quantum phase transitions.
By using quantum renormalization group (QRG) approach, we first derive the effective Hamiltonian and QRG equations of the two-dimensional (2D) Ising models with two different time-dependent transverse magnetic fields analytically. Then we examine the nonanalytic and scaling behaviors of the linear-entropy-based uncertainty relation and quantum entanglement of the models near the critical point through numerical analysis. Moreover, we investigate the relation between the quantum critical point and the external magnetic field. Our results show that both the uncertainty relation and the quantum entanglement are feasible to detect the quantum phase transition (QPT), and the uncertainty relation may be a better indicator of QPT than quantum entanglement. Our findings could shed new light on the observable of the QPTs of the solid-state system with the uncertainty relation.
Uncertainty principle establishes a remarkable lower bound to predict the measured outcome of two non-commuting observables. In this paper, based on the quantum renormalization-group method, we study the relation between quantum-memory-assisted entropic uncertainty relation (QMA-EUR) and quantum phase transition (QPT) in the spin XXZ model. The results shows that the entropic uncertainty and the lower bound have similar traits. In addition, we propose two schemes, one is based on quantum discord and classical correlation, the other Holevo quantity and mutual information, both can tighten the bound of EUR in the presence of quantum memory. The tighter the entropy uncertainty relationship is, the higher the accuracy of the predicted results will be. Moreover, we can obtain the optimal lower bound with the help of Holevo quantity and mutual information, which have the best optimization effect in this model. Additionally, we study QPT by virtue of EUR and after a certain number of iterations, finding that the value of QMA-EUR of the whole block-block state can form two saturated values, which are related to two different phases: spin-fluid phase and Néel phase. Afterwards, we discover that the QMA-EUR of the block-block state obeys the nonanalytic and scaling properties with entropic uncertainty relation exponent associated with correlation length. Our findings show that QMA-EUR deserves to be used as an effective tool in reflecting quantum criticality for more quantum many-body systems and may also shed light on many applications in quantum physics including the quantum key distribution, the detection of QPT and the evaluation of the capacity of quantum computation in critical systems.
Using the quantum renormalization group (QRG) method, we investigate the ground state geometric measure of quantum discord (GQD) of the transverse Ising model. The analytical expression of GQD together with the RG recurrence relations are obtained. The critical and scaling behaviors of GQD are investigated. Our results show that with the increasing of RG steps, GQD (first derivative of GQD) exhibits sudden change (dip) in behavior at the critical point, which indicates that the system undergoes phase transitions between the Ising and the paramagnetic phases. The critical exponent that governs the divergences of the first derivative of GQD is calculated. How the critical point of the model is approached as the increasing of the system size is revealed.
Abstract In this paper, we investigate the relation among local quantum coherence, quantum uncertainty, and the nonlocal advantage of quantum coherence based on skew information and quantum phase transition in the transverse Ising model by exploring the quantum renormalization group (QRG) method. The results reveal that the amount of the local quantum uncertainty is equal to the local quantum coherence corresponding to the local observable σ z in the model, which can be generalized to a multipartite system. Moreover, the nonlocal advantage of quantum coherence is investigated, and we found that regardless of the value of the external magnet field and the number of QRG iterations, the quantum coherence of the subsystem was steerable, which is not only suitable for the two sites of the block, but also for the nearest-neighbor blocks in the long-ranged ferromagnetic phase. However, as the system becomes large enough, the quantum coherence of the subsystem is not steerable in the paramagnetic phase. Additionally, the QRG implementation of quantum coherence and uncertainty are effective and feasible to detect the quantum critical points associated with quantum phase transitions. We also make use of the QRG method to analyze the thermodynamic limit of the current model and the emergence of the nonanalytic and scaling behaviors of the nonlocal advantage of quantum coherence.
Abstract We investigate the quantum coherence behavior of the ground states of 2D Heisenberg XY model and 2D Ising model with a transverse field on square lattices using the Quantum Renormalization Group (QRG) method. Our analysis focused on the ground state density matrix and its marginal states, revealing non-analytic behavior of quantum coherence (especially two-site coherence) near the critical point. This behavior allowed us to detect quantum phase transitions (QPT) in these models. By examining the scaling behavior of the maximum derivative of quantum coherence with system size, we determined the critical exponent of coherence for both models and the length exponent of the Ising model. Additionally, we investigated the time evolution of coherence in both models. Our results closely align with those obtained from entanglement analysis, that is while quantum coherence requires fewer computational calculations compared to discord and entanglement approaches.