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Physica Scripta

Entanglement dynamics in systems of qubits with Markov environments

Björn Lyttkens Lindén, Nikola Burić

Published Jul 1, 2009• 0 citations
Publisher

Abstract

Entanglement dynamics in representative examples of Markov open quantum systems with qualitatively different dynamics are studied. Rings of qubits with thermal or dephasing local environment are used to study the qualitative properties of the entanglement dynamics depending on the interqubit interaction, type of environment and the initial state. It is demonstrated that the effect of the local environment is manifested as an exponential decrease of the entanglement superimposed on the entanglement dynamics in the isolated system.

DOI
10.1088/0031-8949/2009/t135/014051

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Entanglement dynamics in representative examples of Markov open quantum systems with qualitatively different dynamics are studied. Rings of qubits with thermal or dephasing local environment are used to study the qualitative properties of the entanglement dynamics depending on the interqubit interaction, type of environment and the initial state. It is demonstrated that the effect of the local environment is manifested as an exponential decrease of the entanglement superimposed on the entanglement dynamics in the isolated system.
Numerical solutions of the stochastic Schr\"odinger equation given by quantum state diffusion approach to open quantum systems is used to study dynamics of nearest neighbor qubit pairs in systems of small number of qubits on rings with Heisenberg and transverse Ising interaction and under the influence of the thermal environment. In particular, the dependence of the pair entanglement dynamics on the temperature, number of qubits, the type of coupling, and the type of entanglement in the initial state was analyzed for systems of up to N=10N=10N=10 qubits. Periodic recurrence of relatively large values of the pair entanglement with dumping due to decoherence by the thermal noise is observed. It is concluded that the pair entanglement in rings with transverse Ising coupling and prepared in a separable initial state is the most resistant on the decoherence effects of the thermal noise, compared to the Heisenberg coupling or initial states with different types of entanglement.
It is shown that dynamical entanglement between two qubits depends on the symmetry of the quantum model. On the other hand, the latter is reflected in the qualitative properties of the dynamics of a classical approximation of the quantum system. For generic separable pure initial states, the dynamical entanglement is larger if the system is less symmetric and its classical approximation is chaotic. The influence of different types of Markov environments on the established relation between the dynamical entanglement, symmetry and the classical dynamics is also studied.
environment on the entanglement dynamics in non-Markov process. Our results show that the entanglement degree is strongly related with the initial quantum state and the coupling between the qubits and the environment. The interaction between the qubits and the environment suppresses the decay of the entanglement. The sudden death of entanglement (ESD) can be prevented only by using and controlling targeted the interaction model between the different qubits and the environment. We also find the dissipation phenomenon always exists when the qubits system is initially in the maximally entangled mixed states.
In general, quantum systems can be divided into two groups of closed quantum systems and open quantum systems. Also open quantum systems are divided into two categories of Markov systems (non-memory) and non-Markov systems (memory). To distinguish these two behaviors, we use of the entanglement dynamic which can provide a lot of information about the evolution of a system. Returning information from environment to system can be translate as increasing of entanglement in specific period of times, so if increase of entanglement seen, system will have non-Markov treatment and if there is uniform reduction in entanglement, there will be Markov treatment. In special problems which have special matrixes, we have to calculate entanglement every time. For this purpose, we will use of incidence that is a scale for entanglement. For qubit systems the formation of incidence is close which called entanglement of formation. Numerical computation is done by MATLAB and FORTRAN and entanglement dynamics varying by time will be gained. Then analyzing of diagrams is done and the type of process such as Markov and non-Markov is recognized. MENFIT that can show treatment of entanglement in every time, is used too.
We investigate system-environment correlations based on the exact dynamics of a qubit and its environment in the framework of pure decoherence (phase damping). We focus on the relation of decoherence and the buildup of system-reservoir entanglement for an arbitrary (possibly mixed) initial qubit state. In the commonly employed regime where the qubit dynamics can be described by a Markov master equation of the Lindblad type, we find that for almost all qubit initial states inside the Bloch sphere, decoherence is complete while the total state is still separable---no entanglement is involved. In general, both ``separable'' and ``entangling'' decoherence occurs, depending on the temperature and initial qubit state. Moreover, we find situations where classical and quantum correlations periodically alternate as a function of time in the regime of low temperatures.
In this paper, we provide an analytical investigation of the entanglement dynamics of moving qubits dissipating into a common and (in general) non-Markovian environment for both weak and strong coupling regimes. We first consider the case of two moving qubits in a common environment and then generalize it to an arbitrary number of moving qubits. Our results show that when the system evolves from an initial entangled state, the amount of initial entanglement decreases and finally disappears after a finite interval of time due to the environmental effects. Moreover, we observe that the movement of qubits has a constructive role in the protection of the initial entanglement. In a sense, in this case, we observe a Zeno-like effect due to the velocity of qubits. On the other hand, we demonstrate how a stationary state of entanglement may be achieved when we consider the case in which at least one of the moving qubits is initially in the ground state. Surprisingly, we observe that when we extend the number of moving qubits with the same velocity, the stationary state of the qubits does not depend on the velocity of qubits as well as on the environmental properties. This means that, in this condition, the stationary state of entanglement depends only on the number of moving qubits.