We consider two qubits interacting with local and collective thermal reservoirs. Each spin-reservoir interaction consists of an energy exchange and an energy conserving channel. We prove a resonance representation of the reduced dynamics of the spins, valid for all times t ⩾ 0, with errors (small interaction) estimated rigorously, uniformly in time. Subspaces associated to non-interacting energy differences evolve independently, partitioning the reduced density matrix into dynamically decoupled clusters of jointly evolving matrix elements. Within each subspace the dynamics is Markovian with a generator determined entirely by the resonance data of the full Hamiltonian. Based on the resonance representation we examine the evolution of entanglement (concurrence). We show that, whenever thermalization takes place, entanglement of any initial state dies out in a finite time and will not return. For a concrete class of initially entangled spin states we find explicit bounds on entanglement survival and death times in terms of the initial state and the resonance data.We consider two qubits interacting with local and collective thermal reservoirs. Each spin-reservoir interaction consists of an energy exchange and an energy conserving channel. We prove a resonance representation of the reduced dynamics of the spins, valid for all times t ⩾ 0, with errors (small interaction) estimated rigorously, uniformly in time. Subspaces associated to non-interacting energy differences evolve independently, partitioning the reduced density matrix into dynamically decoupled clusters of jointly evolving matrix elements. Within each subspace the dynamics is Markovian with a generator determined entirely by the resonance data of the full Hamiltonian. Based on the resonance representation we examine the evolution of entanglement (concurrence). We show that, whenever thermalization takes place, entanglement of any initial state dies out in a finite time and will not return. For a concrete class of initially entangled spin states we find explicit bounds on entanglement survival and death times in terms of the initial state and the resonance data.

Aug 17, 201610 citations

Evolution of entanglement

Marco Merkli, G. P. Berman, F. Borgonovi +1 more

We analyze the dynamics of entanglement between two qubits which interact through collective and local environments. Our approach is based on a resonance theory which assumes a small interaction between qubits and environments and which gives rigorous perturbation theory results, valid for all times. We obtain expressions for (i) characteristic time-scales for decoherence, relaxation, disentanglement, and for (ii) the evolution of observables, valid uniformly in time $t\geq 0$ . We introduce a classification of decoherence times based on clustering of the reduced density matrix elements, persisting on all time-scales. We examine characteristic dynamical properties such as creation, death and revival of entanglement. We discuss possible applications of our results for superconducting quantum computation and quantum measurement technologies.We analyze the dynamics of entanglement between two qubits which interact through collective and local environments. Our approach is based on a resonance theory which assumes a small interaction between qubits and environments and which gives rigorous perturbation theory results, valid for all times. We obtain expressions for (i) characteristic time-scales for decoherence, relaxation, disentanglement, and for (ii) the evolution of observables, valid uniformly in time $t\geq 0$. We introduce a classification of decoherence times based on clustering of the reduced density matrix elements, persisting on all time-scales. We examine characteristic dynamical properties such as creation, death and revival of entanglement. We discuss possible applications of our results for superconducting quantum computation and quantum measurement technologies.

Quantum Information and ComputationMay 1, 20111 citation

Evolution of Entanglement of Two Qubits Interacting through Local and Collective Environments

Marco Merkli, G. P. Berman, F. Borgonovi +1 more

We analyze rigorously the dynamics of the entanglement between two qubits which interact only through collective and local environments. Our approach is based on the resonance perturbation theory which assumes a small interaction between the qubits and the environments. The main advantage of our approach is that the expressions for (i) characteristic time-scales, such as decoherence, disentanglement, and relaxation, and (ii) observables are not limited by finite times. We introduce a new classification of decoherence times based on clustering of the reduced density matrix elements. The characteristic dynamical properties such as creation and decay of entanglement are examined. We also discuss possible applications of our results for superconducting quantum computation and quantum measurement technologies.We analyze rigorously the dynamics of the entanglement between two qubits which interact only through collective and local environments. Our approach is based on the resonance perturbation theory which assumes a small interaction between the qubits and the environments. The main advantage of our approach is that the expressions for (i) characteristic time-scales, such as decoherence, disentanglement, and relaxation, and (ii) observables are not limited by finite times. We introduce a new classification of decoherence times based on clustering of the reduced density matrix elements. The characteristic dynamical properties such as creation and decay of entanglement are examined. We also discuss possible applications of our results for superconducting quantum computation and quantum measurement technologies.

arXiv:1001.1144Jan 7, 20107 citations

Entanglement in an open Heisenberg spin chain

Subhendu Paul

We analyze the time evolution of a spin chain of three qubits with nearest-neighbour Heisenberg XX interaction, coupled to two thermal reservoirs. We use the ‘reduced density matrix approach’ to calculate the populations and correlations (concurrence) among the spins. The spin-reservoir interaction is of the energy conserving type. This allows us to analyze the evolution analytically without approximations – yielding explicit, yet complicated formulas for the spin dynamics. To analyze the latter and discuss it as a function of the various physical parameters and time (intial conditions, termperatures, coupling strenghts), we resort to numerical methods. Our main finding is that the external noise (coupling to reservoirs) causes the appearence of an asymptotic, stationary (time-periodic) regime for the dynamics of concurrence. We discuss the dependence of the relaxation time and the characteristics of the regime itself, on the various parameters.We analyze the time evolution of a spin chain of three qubits with nearest-neighbour Heisenberg XX interaction, coupled to two thermal reservoirs. We use the ‘reduced density matrix approach’ to calculate the populations and correlations (concurrence) among the spins. The spin-reservoir interaction is of the energy conserving type. This allows us to analyze the evolution analytically without approximations – yielding explicit, yet complicated formulas for the spin dynamics. To analyze the latter and discuss it as a function of the various physical parameters and time (intial conditions, termperatures, coupling strenghts), we resort to numerical methods. Our main finding is that the external noise (coupling to reservoirs) causes the appearence of an asymptotic, stationary (time-periodic) regime for the dynamics of concurrence. We discuss the dependence of the relaxation time and the characteristics of the regime itself, on the various parameters.

Sep 1, 20180 citations

Non-Markovian reduced dynamics and entanglement evolution of two coupled spins in a quantum spin environment

Xiao-Zhong Yuan, Hsi‐Sheng Goan, Ka-Di Zhu

The exact quantum dynamics of the reduced density matrix of two coupled spin qubits in a quantum Heisenberg $XY$ spin star environment in the thermodynamic limit at arbitrarily finite temperatures is obtained using an operator technique. In this approach, the transformed Hamiltonian becomes effectively Jaynes-Cumming-like and thus the analysis is also relevant to cavity quantum electrodynamics. This special operator technique is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. To study their entanglement evolution, the concurrence of the reduced density matrix of the two coupled central spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled central spins, the thermal temperature of the spin environment, and the interaction between the constituents of the spin environment. We also investigate the effect of detuning, which in our model can be controlled by the strength of a locally applied external magnetic field. It is found that the detuning has a significant effect on the entanglement generation between the two-spin qubits.The exact quantum dynamics of the reduced density matrix of two coupled spin qubits in a quantum Heisenberg $XY$ spin star environment in the thermodynamic limit at arbitrarily finite temperatures is obtained using an operator technique. In this approach, the transformed Hamiltonian becomes effectively Jaynes-Cumming-like and thus the analysis is also relevant to cavity quantum electrodynamics. This special operator technique is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. To study their entanglement evolution, the concurrence of the reduced density matrix of the two coupled central spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled central spins, the thermal temperature of the spin environment, and the interaction between the constituents of the spin environment. We also investigate the effect of detuning, which in our model can be controlled by the strength of a locally applied external magnetic field. It is found that the detuning has a significant effect on the entanglement generation between the two-spin qubits.

arXiv:cond-mat/0612049Physical Review BJan 19, 2007100 citations

Dynamics of Collective Decoherence and Thermalization

We analyze the dynamics of N interacting spins (quantum register) collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part. We find the decay rates of the reduced density matrix elements in the energy basis. We show that if the spins do not interact among each other, then the fastest decay rates of off-diagonal matrix elements induced by the energy conserving inter-action is of order N2, while that one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. For a general spin system the decay rates depend in a rather complicated (but explicit) way on the size N and the interaction between the spins. Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings. We do not make Markov-, Born- or weak coupling (van Hove) approximations. 1We analyze the dynamics of N interacting spins (quantum register) collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part. We find the decay rates of the reduced density matrix elements in the energy basis. We show that if the spins do not interact among each other, then the fastest decay rates of off-diagonal matrix elements induced by the energy conserving inter-action is of order N2, while that one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. For a general spin system the decay rates depend in a rather complicated (but explicit) way on the size N and the interaction between the spins. Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings. We do not make Markov-, Born- or weak coupling (van Hove) approximations. 1

Jan 1, 20080 citations

t\geq 0

. We introduce a classification of decoherence times based on clustering of the reduced density matrix elements, persisting on all time-scales. We examine characteristic dynamical properties such as creation, death and revival of entanglement. We discuss possible applications of our results for superconducting quantum computation and quantum measurement technologies.

The exact quantum dynamics of the reduced density matrix of two coupled spin qubits in a quantum Heisenberg

XY

spin star environment in the thermodynamic limit at arbitrarily finite temperatures is obtained using an operator technique. In this approach, the transformed Hamiltonian becomes effectively Jaynes-Cumming-like and thus the analysis is also relevant to cavity quantum electrodynamics. This special operator technique is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. To study their entanglement evolution, the concurrence of the reduced density matrix of the two coupled central spins is also obtained exactly. It is shown that the dynamics of the entanglement depends on the initial state of the system and the coupling strength between the two coupled central spins, the thermal temperature of the spin environment, and the interaction between the constituents of the spin environment. We also investigate the effect of detuning, which in our model can be controlled by the strength of a locally applied external magnetic field. It is found that the detuning has a significant effect on the entanglement generation between the two-spin qubits.