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arXiv:0704.1172 · Physical Review A

Disentanglement in a quantum-critical environment

Zhe Sun, Xiaoguang Wang, C. P. Sun

Published Jun 11, 2007• 130 citations
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Abstract

We study the dynamical process of disentanglement of two qubits and two qutrits coupled to an Ising spin chain in a transverse field, which exhibits a quantum phase transition. We use the concurrence and negativity to quantify entanglement of two qubits and two qutrits, respectively. Explicit connections between the concurrence (negativity) and the decoherence factors are given for two initial states, the pure maximally entangled state and the mixed Werner state. We find that the concurrence and negativity decay exponentially with fourth power of time in the vicinity of the critical point of the environmental system.

DOI
10.1103/physreva.75.062312

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We study the dynamical process of disentanglement of two qubits and two qutrits coupled to an Ising spin chain in a transverse field, which exhibits a quantum phase transition. We use the concurrence and negativity to quantify entanglement of two qubits and two qutrits, respectively. Explicit connections between the concurrence (negativity) and the decoherence factors are given for two initial states, the pure maximally entangled state and the mixed Werner state. We find that the concurrence and negativity decay exponentially with fourth power of time in the vicinity of the critical point of the environmental system.
In this paper, we study the dynamical processes of quantum coherence and correlations for two central qubits system coupled with a transverse Ising spin chain. Suppose the initial state of quantum system is the Werner state and the initial state of environment is the ground state of spin chain, and the corresponding time evolution operator, decoherence factor and reduced density matrix are given. We deduce the analytical expressions of evolution of quantum coherence, entanglement and quantum discord. We find that when the spin chain undergoes quantum phase transition (QPT), the entanglement vanished at time [Formula: see text], and the coherence and discord vanished when the decoherence factor decayed to zero. Besides, we also find that the environmental scale N, coupling strength g and parameter P of Werner state do not change the evolution rules of quantum correlation, but accelerate their decay rate with these parameters’ increase.
We study the disentanglement evolution of two spin qubits which interact with a general XYXYXY spin-chain environment. The dynamical process of the disentanglement is numerically and analytically investigated in the vicinity of a quantum phase transition (QPT) of the spin chain in both weak and strong coupling cases. We find that the disentanglement of the two spin qubits may be greatly enhanced by the quantum critical behavior of the environmental spin chain. We give a detailed analysis to facilitate the understanding of the QPT-enhanced decaying behavior of the coherence factor. Furthermore, the scaling behavior in the disentanglement dynamics is also revealed and analyzed.
We study the dynamics of disentanglement of two qubits initially prepared in a Bell state and coupled at different sites to an Ising spin chain in a transverse field (ITF) playing the role of a dynamic spin environment. The initial state of the whole system is prepared in a tensor product state ${\ensuremath{\rho}}_{\mathrm{Bell}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{\mathrm{chain}}$, where the state of the chain is taken to be given by the ground state $|G({\ensuremath{\lambda}}_{i})\ensuremath{\rangle}$ of the ITF Hamiltonian ${H}_{E}({\ensuremath{\lambda}}_{i})$ with an initial field ${\ensuremath{\lambda}}_{i}$. At time t=0+t={0}^{+}t=0+, the strength of the transverse field is suddenly quenched to a new value, ${\ensuremath{\lambda}}_{f}$, and the whole system (chain +++ qubits) undergoes a unitary dynamics generated by the total Hamiltonian ${H}_{\mathrm{Tot}}={H}_{E}({\ensuremath{\lambda}}_{f})+{H}_{I}$, where HI{H}_{I}HI​ describes a local interaction between the qubits and the spin chain. The resulting dynamics leads to a disentanglement of the qubits, which is described through Wootters' concurrence, due to their interaction with the nonequilibrium environment. The concurrence is related to the Loschmidt echo, which in turn is expressed in terms of the time-dependent covariance matrix associated with the ITF. This permits a precise numerical and analytical analysis of the disentanglement dynamics of the qubits as a function of their distance, bath properties, and quench amplitude. In particular, we emphasize the special role played by a critical initial environment.
We consider a generalized central spin model, consisting of two central qubits and an environmental spin chain (with periodic boundary condition) to which these central qubits are locally and weakly connected either at the same site or at two different sites separated by a distance ddd. Our purpose is to study the subsequent temporal generation of entanglement, quantified by concurrence, when initially the qubits are in an unentangled state. In the equilibrium situation, we show that the concurrence survives for a larger value of ddd when the environmental spin chain is critical. Importantly, a common feature observed both in the equilibrium and the nonequilibrium situations while the latter is created by a sudden but global change of the environmental transverse field is that the two qubits become maximally entangled for the critical quenching. Following a nonequilibrium evolution of the spin chain, our study for $d\ensuremath{\ne}0$ indicates that there exists a threshold time above which concurrence attains a finite value. Additionally, we show that the number of independent decohering channels (DCs) is determined by ddd as well as the local difference of the transverse field of the two underlying Hamiltonians governing the time evolution; the concurrence can be enhanced by a higher number of independent channels. The qualitatively similar behavior displayed by the concurrence for critical and off-critical quenches, as reported here, is characterized by analyzing the nonequilibrium evolution of these channels. The concurrence is maximum when the decoherence factor or the echo associated with the most rapidly DC decays to zero; on the contrary, the condition when the concurrence vanishes is determined nontrivially by the associated decay of one of the intermediate DCs. Analyzing the reduced density of a single qubit, we also explain the observation that the dephasing rate is always slower than the unentanglement rate. We further establish that the maximally and minimally decohering channels show singular behavior which can be explained by invoking upon a quasiparticle picture.
We review recent studies on the measures of zero temperature quantum correlations namely, the quantum entanglement (concurrence) and discord present in the final state of a transverse XY spin chain following a quench through quantum critical points; the aim of these studies is to explore the scaling of the above quantities as a function of the quench rate. A comparative study between the concurrence and the quantum discord shows that their behavior is qualitatively the same though there are quantitative differences. For the present model, the scaling of both the quantities are given by the scaling of the density of the defect present in the final state though one cannot find a closed form expression for the discord. We also extend our study of quantum discord to a transverse Ising chain in the presence of a three spin interaction. Finally, we present a study of the dynamical evolution of quantum discord and concurrence when two central qubits, initially prepared in a Werner state, are coupled to the environmental XY spin chain which is driven through quantum critical points. The qualitative behavior of quantum discord and concurrence are found to be similar as that of the decoherence factor.
We investigate the entanglement dynamics of three-qubit states in a quantum-critical environment, which is an Ising model in a transverse field. By using negativity as entanglement measure, we find that the entanglement evolution depends not only on the system-environment couplings and the size of degrees of freedom of environment but also on the strength of transverse field and the symmetry of quantum states of concern. In particular, for the cases under study, our results imply that the entanglement decay can be enhanced by the quantum phase transition of the environment under weak coupling. Our analysis implies that the entanglement of any three-qubit quantum state will be completely destroyed by the decoherence under certain conditions, that we detail below.