We consider spatially separated qubits coupled to a thermal bosonic field that causes pure dephasing. Our focus is on the entanglement of two Bell states which for vanishing separation are known as robust and fragile entangled states. The reduced two-qubit dynamics is solved exactly and explicitly. Our results allow us to gain information about the robustness of two-qubit decoherence-free subspaces with respect to physical parameters such as temperature, qubit-bath coupling strength and spatial separation of the qubits. Moreover, we clarify the relation between single-qubit coherence and two-qubit entanglement and identify parameter regimes in which the terms robust and fragile are no longer appropriate.We consider spatially separated qubits coupled to a thermal bosonic field that causes pure dephasing. Our focus is on the entanglement of two Bell states which for vanishing separation are known as robust and fragile entangled states. The reduced two-qubit dynamics is solved exactly and explicitly. Our results allow us to gain information about the robustness of two-qubit decoherence-free subspaces with respect to physical parameters such as temperature, qubit-bath coupling strength and spatial separation of the qubits. Moreover, we clarify the relation between single-qubit coherence and two-qubit entanglement and identify parameter regimes in which the terms robust and fragile are no longer appropriate.

Jan 1, 200638 citations

Incomplete pure dephasing of N -qubit entangled W states

Roland Doll, Martijn Wubs, Peter Hänggi A +1 more

We consider qubits in a linear arrangement coupled to a bosonic field which acts as a quantum heat bath and causes decoherence. By taking the spatial separation of the qubits explicitly into account, the reduced qubit dynamics acquires an additional non-Markovian element. We investigate the exact time evolution of an entangled many-qubit W state, which for vanishing qubit separation remains robust under pure dephasing. For finite separation, by contrast, the dynamics is no longer decoherence-free. On the other hand, spatial noise correlations may prevent complete dephasing. While a standard Bloch-Redfield master equation fails to describe this behavior even qualitatively, we propose instead a widely applicable causal master equation. Here we employ it to identify and characterize decoherence-poor subspaces. Consequences for quantum error correction are discussed.We consider qubits in a linear arrangement coupled to a bosonic field which acts as a quantum heat bath and causes decoherence. By taking the spatial separation of the qubits explicitly into account, the reduced qubit dynamics acquires an additional non-Markovian element. We investigate the exact time evolution of an entangled many-qubit W state, which for vanishing qubit separation remains robust under pure dephasing. For finite separation, by contrast, the dynamics is no longer decoherence-free. On the other hand, spatial noise correlations may prevent complete dephasing. While a standard Bloch-Redfield master equation fails to describe this behavior even qualitatively, we propose instead a widely applicable causal master equation. Here we employ it to identify and characterize decoherence-poor subspaces. Consequences for quantum error correction are discussed.

arXiv:cond-mat/0703075Physical Review BJul 17, 200729 citations

Phonon decoherence of quantum entanglement: Robust and fragile states

T. Yu, Joseph H. Eberly

We study the robustness and fragility of entanglement of open quantum systems in some exactly solvable models in which the decoherence is caused by a pure dephasing process. In particular, for the toy models presented in this paper, we identify two different time scales, one is responsible for local dephasing, while the other is for entanglement decay. For a class of fragile entangled states defined in this paper, we find that the entanglement of two qubits, as measured by concurrence, decays faster asymptotically than the quantum dephasing of an individual qubit.We study the robustness and fragility of entanglement of open quantum systems in some exactly solvable models in which the decoherence is caused by a pure dephasing process. In particular, for the toy models presented in this paper, we identify two different time scales, one is responsible for local dephasing, while the other is for entanglement decay. For a class of fragile entangled states defined in this paper, we find that the entanglement of two qubits, as measured by concurrence, decays faster asymptotically than the quantum dephasing of an individual qubit.

arXiv:quant-ph/0209037Physical review. B, Condensed matterNov 13, 2002187 citations

System-environment correlations for dephasing two-qubit states coupled to thermal baths

Ana C. S. Costa, Marcus W. Beims, Walter T. Strunz

Based on the exact dynamics of a two-qubit system and environment, we investigate system-environment (SE) quantum and classical correlations. The coupling is chosen to represent a dephasing channel for one of the qubits and the environment is a proper thermal bath. First we discuss the general issue of dilation for qubit phase damping. Based on the usual thermal bath of harmonic oscillators, we derive criteria of separability and entanglement between an initial $X$ state and the environment. Applying these criteria to initial Werner states, we find that entanglement between the system and environment is built up in time for temperatures below a certain critical temperature ${T}_{\mathrm{crit}}$ . On the other hand, the total state remains separable during those short times that are relevant for decoherence and loss of entanglement in the two-qubit state. Close to ${T}_{\mathrm{crit}}$ the SE correlations oscillate between separable and entangled. Even though these oscillations are also observed in the entanglement between the two qubits, no simple relation between the loss of entanglement in the two-qubit system and the build-up of entanglement between the system and environment is found.Based on the exact dynamics of a two-qubit system and environment, we investigate system-environment (SE) quantum and classical correlations. The coupling is chosen to represent a dephasing channel for one of the qubits and the environment is a proper thermal bath. First we discuss the general issue of dilation for qubit phase damping. Based on the usual thermal bath of harmonic oscillators, we derive criteria of separability and entanglement between an initial $X$ state and the environment. Applying these criteria to initial Werner states, we find that entanglement between the system and environment is built up in time for temperatures below a certain critical temperature ${T}_{\mathrm{crit}}$. On the other hand, the total state remains separable during those short times that are relevant for decoherence and loss of entanglement in the two-qubit state. Close to ${T}_{\mathrm{crit}}$ the SE correlations oscillate between separable and entangled. Even though these oscillations are also observed in the entanglement between the two qubits, no simple relation between the loss of entanglement in the two-qubit system and the build-up of entanglement between the system and environment is found.

arXiv:1605.04154Physical Review AMay 12, 201615 citations

Controllable dynamics of two separate qubits in Bell states

Jun Jing, Zhiguo Lü, Guo-Hong Yang

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.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.

arXiv:0705.3276Physical Review ASep 21, 200716 citations

Exact dynamics of interacting qubits in a thermal environment: results beyond the weak coupling limit

Lian-Ao Wu, Claire X. Yu, Dvira Segal

We demonstrate an exact mapping of a class of models of two interacting qubits in thermal reservoirs to two separate problems of spin–boson-type systems. Based on this mapping, exact numerical simulations of the qubits dynamics can be performed, beyond the weak system–bath coupling limit and the Markovian approximation. Given the time evolution of the system population and coherences, we study as an application the dynamics of entanglement between the pair of qubits immersed in boson thermal baths, showing a rich phenomenology, including an intermediate oscillatory behavior, the entanglement sudden birth, sudden death and revival. We find that the occurrence of entanglement sudden death in this model depends on the portion of the zero and double excitation states in the subsystem initial state. In the long-time limit, analytic expressions are presented at weak system–bath coupling, for a range of relevant qubit parameters.We demonstrate an exact mapping of a class of models of two interacting qubits in thermal reservoirs to two separate problems of spin–boson-type systems. Based on this mapping, exact numerical simulations of the qubits dynamics can be performed, beyond the weak system–bath coupling limit and the Markovian approximation. Given the time evolution of the system population and coherences, we study as an application the dynamics of entanglement between the pair of qubits immersed in boson thermal baths, showing a rich phenomenology, including an intermediate oscillatory behavior, the entanglement sudden birth, sudden death and revival. We find that the occurrence of entanglement sudden death in this model depends on the portion of the zero and double excitation states in the subsystem initial state. In the long-time limit, analytic expressions are presented at weak system–bath coupling, for a range of relevant qubit parameters.

arXiv:1207.6995New Journal of PhysicsFeb 27, 201312 citations

X

state and the environment. Applying these criteria to initial Werner states, we find that entanglement between the system and environment is built up in time for temperatures below a certain critical temperature

{T}_{\mathrm{crit}}

. On the other hand, the total state remains separable during those short times that are relevant for decoherence and loss of entanglement in the two-qubit state. Close to

{T}_{\mathrm{crit}}

the SE correlations oscillate between separable and entangled. Even though these oscillations are also observed in the entanglement between the two qubits, no simple relation between the loss of entanglement in the two-qubit system and the build-up of entanglement between the system and environment is found.

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.