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Entanglement distribution and its invariance

Qingjun Tong, Hong‐Gang Luo, Changhun Oh

Published May 6, 2010• 0 citations
OpenAlex

Abstract

We study the entanglement dynamics of a two-qubit system, each of which is embedded into its local reservoir, and the entanglement distribution among all the bipartite subsystems including qubit-qubit, qubit-reservoir, and reservoir-reservoir. It is found that the entanglement can be stably distributed among all components and particularly it also satisfies an identity firstly given by Yonac, Yu and Eberly (J. Phys. B 40, S45 (2007)) for a double J-C model without decoherence. While the explicit distribution of the entanglement is dependent of the detail of the model, even the approximation used, the identity remains unchanged. Our unified treatment includes the previous results in the literature as special cases. The result reveals the profound nature of the entanglement and should have significant implications for quantum information processing. below) has been satisfied at any time. This may be not surprising since the double J-C model has no decoherence and any initial information can be preserved in the time evolution. However, it would be surprising if the identity is still valid in the presence of the decoherence, in which a non-equilibrium relaxation process is involved. In this paper, we show that it is indeed true for such a system consisted of two qubits locally interacting with two reser- voirs. We find that the distribution of the entanglement among the bipartite subsystems is dependent of the ex- plicit property of the environment and its coupling with the qubit. The rich dynamical behaviors obtained pre- viously in the literature can be regarded as the special cases of our present result or Markovian approximation. Particularly, we find that the entanglement can stably distribute among all the bipartite subsystems if the qubit and its environment can form a bound state and the non- Markovian effect is important. Irrespective of how dis- tributes the entanglement, it is found that the identity about the entanglement in the whole system can be sat- isfied at any time, which reveals the profound physics of the entanglement dynamics. We consider two qubits interacting with two uncorre- lated vacuum reservoirs. Due to the dynamical indepen- dence between the two local subsystems, we can firstly solve the single subsystem, then apply the result obtained to the double-qubit case. The Hamiltonian of each local subsystem is (16)

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We study the entanglement dynamics of a two-qubit system, each of which is embedded into its local reservoir, and the entanglement distribution among all the bipartite subsystems including qubit-qubit, qubit-reservoir, and reservoir-reservoir. It is found that the entanglement can be stably distributed among all components and particularly it also satisfies an identity firstly given by Yonac, Yu and Eberly (J. Phys. B 40, S45 (2007)) for a double J-C model without decoherence. While the explicit distribution of the entanglement is dependent of the detail of the model, even the approximation used, the identity remains unchanged. Our unified treatment includes the previous results in the literature as special cases. The result reveals the profound nature of the entanglement and should have significant implications for quantum information processing. below) has been satisfied at any time. This may be not surprising since the double J-C model has no decoherence and any initial information can be preserved in the time evolution. However, it would be surprising if the identity is still valid in the presence of the decoherence, in which a non-equilibrium relaxation process is involved. In this paper, we show that it is indeed true for such a system consisted of two qubits locally interacting with two reser- voirs. We find that the distribution of the entanglement among the bipartite subsystems is dependent of the ex- plicit property of the environment and its coupling with the qubit. The rich dynamical behaviors obtained pre- viously in the literature can be regarded as the special cases of our present result or Markovian approximation. Particularly, we find that the entanglement can stably distribute among all the bipartite subsystems if the qubit and its environment can form a bound state and the non- Markovian effect is important. Irrespective of how dis- tributes the entanglement, it is found that the identity about the entanglement in the whole system can be sat- isfied at any time, which reveals the profound physics of the entanglement dynamics. We consider two qubits interacting with two uncorre- lated vacuum reservoirs. Due to the dynamical indepen- dence between the two local subsystems, we can firstly solve the single subsystem, then apply the result obtained to the double-qubit case. The Hamiltonian of each local subsystem is (16)
We study the entanglement dynamics of two qubits, each of which is embedded into its local amplitude-damping reservoir, and the entanglement distribution among all the bipartite subsystems including qubit-qubit, qubit-reservoir, and reservoir-reservoir. It is found that the entanglement can be stably distributed among all components, which is much different to the result obtained under the Born-Markovian approximation by C. E. López {\it et al.} [Phys. Rev. Lett. \textbf{101}, 080503 (2008)], and particularly it also satisfies an identity. Our unified treatment includes the previous results as special cases. The result may give help to understand the physical nature of entanglement under decoherence.
We study the entanglement dynamics of two qubits, each of which is embedded into its local amplitude-damping reservoir, and the entanglement distribution among all the bipartite subsystems including qubit-qubit, qubit-reservoir, and reservoir-reservoir. It is found that the entanglement can be stably distributed among all components, which is much different to the result obtained under the Born-Markovian approximation by C. E. L\'{o}pez et al., and particularly it also satisfies an identity. Our unified treatment includes the previous results as special cases. The result may give help to understand the physical nature of entanglement under decoherence.
The thesis is contributed to the study of the decoherence dynamics of dissipative qubit systems. We reveal the profound impact of the formation of a bound state between the qubit and its local environment on the decoherence dynamics of qubit system in non-Markovian regime. We find that the decoherence can be suppressed greatly under the non-Markovian dynamics when the bound state is formed. Consequently, the entanglement can be partially preserved in the steady state. A criterion on this decoherence/disentanglement suppression is given explicitly. It can give a useful guideline on the decoherence control in experiment. We also find that the entanglement can be stably distributed among all components and particularly it always satisfies an identity. Our unified treatment includes the previous results in the literatures as special cases. The result reveals the profound nature of the entanglement and should have significant implications for quantum information processing.
In this Thesis I discuss the exact dynamics of simple non-Markovian systems. I focus on fundamental questions at the core of non-Markovian theory and investigate the dynamics of quantum correlations under non-Markovian decoherence. \n \nIn the first context I present the connection between two different non-Markovian approaches, and compare two distinct definitions of non-Markovianity. The general aim is to characterize in exemplary cases which part of the environment is responsible for the feedback of information typical of non- Markovian dynamics. I also show how such a feedback of information is not always described by certain types of master equations commonly used to tackle non-Markovian dynamics. \n \nIn the second context I characterize the dynamics of two qubits in a common non-Markovian reservoir, and introduce a new dynamical effect in a wellknown model, i.e., two qubits under depolarizing channels. In the first model the exact solution of the dynamics is found, and the entanglement behavior is extensively studied. The non-Markovianity of the reservoir and reservoirmediated-interaction between the qubits cause non-trivial dynamical features. \n \nThe dynamical interplay between different types of correlations is also investigated. In the second model the study of quantum and classical correlations demonstrates the existence of a new effect: the sudden transition between classical and quantum decoherence. This phenomenon involves the complete preservation of the initial quantum correlations for long intervals of time of the order of the relaxation time of the system.
We study the decoherence properties of an entangled bipartite qubit system, represented by two two-level atoms that are individually coupled to non-Markovian reservoirs. This coupling ensures that the dynamical equations of the atoms can be treated independently. The non-Markovian reservoirs are described by a model which leads to an exact non-Markovian master equation of the Nakajima-Zwanzig form [J. Salo, S. M. Barnett, and S. Stenholm, Optics Commun. 259, 772 (2006)]. We consider the evolution of the entanglement of a two-atom state that is initially completely entangled, quantified by its concurrence. Collapses and revivals in the concurrence, induced by the memory effects of the reservoir, govern the dynamics of the entangled quantum system. These collapses and revivals in the concurrence are a strong manifestation of the non-Markovian reservoir.
We present an approach to the study of entanglement decay which focuses on collective properties. As an example, we investigate the entanglement decay of a two-qubit system produced by local identical reservoirs acting on the qubits for three experimentally and theoretically relevant cases. We study the probability distributions of disentanglement times, a quantity independent of the measure used to quantify entanglement, and the time-dependent probability distribution of concurrence. Analytical results are obtained for initially uniformly distributed pure states. The calculation of these probability distributions gives a complete insight into how different decoherence channels affect the entanglement initially contained in the set of two-qubit pure states. Numerical results are reported for randomly distributed initial mixed states. Although the paper focuses in Markovian noisy channels, we show that our results also describe nonautonomous and non-Markovian channels.