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Physical Review A

Markovian and non-Markovian dynamics of quantum coherence in the extended X X chain

Shaoying Yin, Shutian Liu, Jie Song, Hongliang Luan

Published Sep 27, 2022• 10 citations
Publisher

Abstract

The Markovian and non-Markovian dynamics of quantum coherence, including single-spin coherence, two-spin coherence, and its distribution (collective and localized coherence), are investigated in the $XX$ spin chain with three-spin interaction. We find that the single-spin and localized coherence do not decrease but increase with time, which makes the dynamic evolution of the two-spin coherence more robust than entanglement and quantum discord. The three-spin interaction can lower the values of quantum coherence in Markovian evolution and heighten the oscillation frequency of quantum coherence in non-Markovian evolution because it can speed up the flow of coherence information. Finally, the trade-off relation between the collective and the localized coherence is clearly shown in their dynamic evolution. Thanks to the coherence trade-off, we find the substitution of localized coherence for collective coherence in dominating the dynamics of the two-spin coherence.

DOI
10.1103/physreva.106.032220

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The Markovian and non-Markovian dynamics of quantum coherence, including single-spin coherence, two-spin coherence, and its distribution (collective and localized coherence), are investigated in the XXXXXX spin chain with three-spin interaction. We find that the single-spin and localized coherence do not decrease but increase with time, which makes the dynamic evolution of the two-spin coherence more robust than entanglement and quantum discord. The three-spin interaction can lower the values of quantum coherence in Markovian evolution and heighten the oscillation frequency of quantum coherence in non-Markovian evolution because it can speed up the flow of coherence information. Finally, the trade-off relation between the collective and the localized coherence is clearly shown in their dynamic evolution. Thanks to the coherence trade-off, we find the substitution of localized coherence for collective coherence in dominating the dynamics of the two-spin coherence.
We study the dynamics of entanglement, mutual information, and quantum discord in the extended cluster spin-1/2 XX\mathit{XX}XX chain, equivalent to a one-dimensional spin-1/2 XX\mathit{XX}XX model with three-spin interaction (TSI). Selecting the nearest neighbor pair spins as an open quantum system, the rest of the chain plays the role of the environment. The two-point Heisenberg and the TSI are responsible for coupling between the system and the environment. Although the revival phenomenon of quantum correlations as an indication of non-Markovian dynamics is observed for TSI stronger than the Heisenberg interaction, the study of the trace distance has proven that the dynamical phase transition from the Markovian to the non-Markovian regime happens at a critical value where the TSI is equal to half of the Heisenberg interaction. By focusing on the nearest neighbor pair spins of the environment, we have also shown that the dynamics of quantum correlation in the environment is sensitive to Markovian and non-Markovian regions.
The behavior of quantum coherence is studied under Markovian and non-Markovian dynamics for open quantum systems. For single qubit systems, we show that the coherence depending on the off-diagonal elements of the density matrix is the upper bound of the coherence depending on the relative entropy under both Markovian and non-Markovian processes. For two-qubit systems, in both Markovian and non-Markovian processes, quantum discord and coherence show less sensitivity to the initial state than quantum entanglement. We also find that the quantum discord has similar behaviors with coherence under both Markovian and non-Markovian dynamics.
One-spin and two-spin in a thermodynamic limit spin-1/2 chain as open quantum systems are considered. The dynamics of system is generated by XX Heisenberg interaction and three-spin interaction (TSI) among all the nearest three spins in the chain. Two variety Hamiltonians of the three-spin interaction are considered. Using the fermionization technique and calculating the trace distance, non-Markovianity as a function of the TSI is evaluated, and the results for the two different open quantum systems are compared. In addition, the time behavior of the probability amplitudes are studied. The results show that if all probability amplitudes except one of them will almost vanish after some time, the dynamics of the open quantum system will be Markovian. In the non-Markovian dynamics, more than one of the probability amplitudes fluctuate in time and will never reach zero value.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the 'spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the `spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.
Quantum coherence has played a decisive role in quantum information processing. On the other hand, quantum correlation can be considered as a powerful resource for delivering quantum information. Both quantum coherence and quantum correlation may occur in an information propagating process, which challenges us to understand the relationship between coherence and correlation. This is also an important procedure for physicists to know the features of quantum resources. Any quantum system interacting with its surrounding environment will destroy the quantum coherence and fail to fulfil any task of delivering quantum information. In this sense, studying the dynamics of quantum correlation and quantum coherence is very fascinating. In this paper, we investigate the dynamics of the quantum correlation and quantum coherence for two central qubits coupled to their own spin baths modeled by the XY spin chain with Dzyaloshinsky-Moriya interaction. We employ the quantum discord to characterize the quantum correlation, and use the relative entropy to measure quantum coherence. In this way the evolution law of the quantum discord and the relative entropy of quantum coherence of two-qubit system are derived, and the evolution law depends not only on the Dzyaloshinsky-Moriya interaction, the anisotropy parameter and the total number of spin chain sites, but also on the coupling strength between the central spin and its spin chain. Our findings are as follows. Firstly, we find that near the critical point of spin chain the quantum coherence abruptly changes, which can be used to detect the existence of quantum phase transition. Secondly, at the critical point, the relative entropy of quantum coherence is the same as that of classical correlation when time tt0, and it is the same as that of quantum discord when time tt0. At time t0, the sudden transition from quantum discord to classical correlation occurs. All in all, the relative entropy of quantum coherence reflects the behaviors of classical correlation and quantum discord for times tt0 and tt0, respectively, which is caused by the change of the optimal basis for quantum discord. Thirdly, the dynamics of quantum correlation and quantum coherence keep invariant under the scaling variation of the total number of spin chain sites and the coupling strength. Moreover, we find that all the Dzyaloshinsky-Moriya interactions and the anisotropy parameters, as well as the coupling strengths will enhance the decay of quantum coherence and quantum correlation, while they have no obvious effect on the relationship between dynamics of coherence and correlation. The above discussion reveals some new features of quantum coherence and quantum correlation, which may be useful in further developing quantum information theory.