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arXiv:0705.3276 · Physical Review A

Controllable dynamics of two separate qubits in Bell states

Jun Jing, Zhiguo Lü, Guo-Hong Yang

Published Sep 21, 2007• 16 citations
arXivHTMLPDF

Abstract

The dynamics of entanglement and fidelity for a subsystem of two separate spin-$1∕2$ qubits prepared in Bell states is investigated. One of the subsystem qubit labeled $A$ is under the influence of a Heisenberg $XY$ spin bath, while another one labeled $B$ is uncoupled with that. We discuss two cases: (i) the number of bath spins is infinite, $N\ensuremath{\rightarrow}\ensuremath{\infty}$, and (ii) $N$ is finite, $N=40$. In both cases, the bath is initially prepared in a thermal equilibrium state. It is shown that the time dependence of the concurrence and the fidelity of the two subsystem qubits can be controlled by tuning the parameters of the spin bath, such as the anisotropic parameter, the temperature, and the coupling strength with qubit $A$. It is interesting to find that the dynamics of the concurrence is independent of four different initial Bell states and that of the fidelity is divided into two groups.

DOI
10.1103/physreva.76.032322

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The dynamics of two 1∕21∕21∕2-spin qubits under the influence of a quantum Heisenberg XYXYXY-type spin-bath is studied. After the Holstein-Primakoff transformation, numerical polynomial scheme is used to give the time-evolution calculation of the center qubits initially prepared in a product state or a Bell state. Then the concurrence of the two qubits, the zzz-component moment of either of the subsystem spins, and the fidelity of the subsystem are shown; they exhibit sensitive dependence on the anisotropic parameter, the temperature, the coupling strength, and the initial state. It is found that (i) the larger the anisotropic parameter $\ensuremath{\gamma}$, the bigger the probability of maintaining the initial state of the two qubits; (ii) with increasing temperature TTT, the bath plays a stronger destroy effect on the dynamics of the subsystem, so does the interaction g0{g}_{0}g0​ between the subsystem and the bath; and (iii) the time evolution of the subsystem is dependent on the initial state. The revival of the concurrence does not always mean the restoration of the state. Further, the dynamical properties of the subsystem should be judged by the combination of concurrence and fidelity.
The dynamics of two coupled spins of 1/2 coupled to spin-bath of a quantum Heisenberg XY type \\cite{Breuer, Yuan} is studied. The center pair of spins served as an quantum open subsystem were initially prepared in a Bell state or a product state and the bath consisted of NNN (NtoinftyN\\to\\inftyNtoinfty as the thermodynamic limit) spins-1/2 is in a thermal state at different temperatures from the beginning. Transformed by the Holstein-Primakoff operator, the model will be treated effectively as two spin qubits embedded in a single mode cavity. Then the von-Neumann entropy, z-component summation and the concurrence of the center spins can be determined by a novel polynomial scheme for the time-evolution of quantum systems. It is found that (i) with increasing temperature, the bath plays a more strong destroy effect on the coherence or entanglement of the subsystem; (ii) the larger the coupling strength between the subsystem spins, the less the variation of the initial state; (iii) the stronger the interaction between the subsystem and the bath, the faster the variation of the initial state.
The dynamics of two coupled spins-1/2 interacting with a spin-bath via the quantum Heisenberg XY coupling is studied. The pair of central spins served as a quantum open subsystem are initially prepared in two types of states: the product states and the Bell states. The bath, which consists of NNN (in the thermodynamic limit N→∞N\to\inftyN→∞) mutually coupled spins-1/2, is in a thermal state at the beginning. By the Holstein-Primakoff transformation, the model can be treated effectively as two spin qubits embedded in a single mode cavity. The time-evolution of the purity, z-component summation and the concurrence of the central spins can be determined by a Laguerre polynomial scheme. It is found that (i) at a low temperature, the uncoupled subsystem in a product state can be entangled due to the interaction with bath, which is tested by the Peres-Horodecki separability; however, at a high temperature, the bath produces a stronger destroy effect on the purity and entanglement of the subsystem; (ii) when the coupling strength between the two central spins is large, they are protected strongly against the bath; (iii) when the interaction between the subsystem and the bath is strong, the collapse of the two spin qubits from their initial entangled state is fast.
The decoherence and entanglement dynamics of two interacting qubits coupled through Heisenberg XY interactions to a spin bath in thermal equilibrium are studied. The exact form of the reduced density matrix is derived for finite and infinite numbers of environmental spins. It is shown that decoherence can be minimized at low bath temperatures and strong coupling between the qubits. Some initial product states evolve into entangled ones, initially entangled states lose completely or partially their entanglement. The relation between the fidelity and the concurrence is also investigated.
Two noninteracting qubits, initially prepared in an entangled state, are coupled to their own independent environments and evolve under their influence. The reduced non-Markovian dynamics of two qubits is exact for arbitrary model parameters. Necessary and sufficient conditions for nonvanishing entanglement are formulated for both zero and nonzero temperatures and arbitrary time. It is shown that (i) entanglement dynamics can effectively be controlled by a finite quantum system coupled to one of the qubits and (ii) dynamical symmetry of the controlling quantum system can influence significantly entanglement of the qubits and results in its nonmonotonic decay.
An exactly solvable model for the decoherence of one and two-qubit states interacting with a spin bath, in the presence of a time-dependent magnetic field is studied. The magnetic field is static along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction and oscillatory in the transverse plane. The transition probability and Rabi oscillations between the spin states of a single qubit is shown to depend on the size of bath, the distribution of qubit-bath couplings, and the initial bath polarization. In contrast to the fast Gaussian decay for short times, the polarization of the qubit shows an oscillatory power-law decay for long times. The loss of entanglement for the maximally entangled two-qubit states can be controlled by tuning the frequency of the rotating field. The decay rates of entanglement and purity for all the Bell states are the same when the qubits are noninteracting, and different when they are interacting.