Incomplete pure dephasing of $N$-qubit entangled W states

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.

Figure 1

(Color online) Schematic representation of $N$ qubits in a linear arrangement. The qubits (green boxes) interact via a coupling to the substrate phonon field (red line) at their positions $x_{\nu }$.

Figure 2

(Color online) Time evolution of the concurrence $C$ for two qubits initially prepared in the robust state (18) for various transit times $t_{12}$: exact time evolution (solid lines) compared to the results obtained from the causal master equation derived in Sec. 4 (dashed). The temperature is $k_{B}T={10}^{-4}\hslash {\omega }_c$ and the coupling strength $\alpha =0.005$. For $\Omega ={10}^{-3}{\omega }_c$, the time range corresponds to 40 coherent oscillations. Inset: Blow-up on a logarithmic scale for the transit time $t_{12}=5\times {10}^4∕{\omega }_c$.

Figure 3

(Color online) Final value of concurrence for the robust state $\mid {\mathrm{W}}_2⟩$ as a function of the spatial separation $x_{12}=c{t}_{12}$ for various temperatures. The coupling strength is $\alpha =0.005$.

Figure 4

(Color online) Exact time evolution of the fidelity $F\left(t\right)$ (solid lines) and the result obtained from the causal master equation (dashed lines) for $N=3$ and 6 qubits, respectively. The temperatures are $k_{B}T={10}^{-4}∕\hslash {\omega }_c$ (a) and $k_{B}T={10}^{-3}∕\hslash {\omega }_c$ (b). As in Fig. 2, the transit time between nearest-neighbor qubits is $t_{12}=5\times {10}^4∕{\omega }_c$ and the qubit-field coupling strength $\alpha =0.005$. The inset in panel (b) shows the data for $N=3$ on a logarithmic time axis.

Figure 5

(Color online) Final fidelity (23) as a function of the number $N$ of qubits at zero temperature for two coupling strengths $\alpha =0.001$ (blue circles) and $\alpha =0.01$ (red diamonds). The nearest-neighbor transit time is again ${\omega }_c{t}_{12}=5\times {10}^4$. The crosses (orange) and plus signs (green) mark the approximate result (25), respectively. The dashed line indicates the lower bound $1∕N$ of the fidelity.

Figure 6

(Color online) Difference between the exact result and the causal master equation result (41) for the final value of the concurrence $\delta C=C\left(\infty \right)-C_{\mathrm{CME}}\left(\infty \right)$ for the robust entangled two-qubit state (18). The coupling strength is $\alpha =0.005$. The inset depicts $\delta C$ for the fixed temperatures $\theta =k_{B}T∕\hslash {\omega }_c={10}^{-4}$ (solid), $\theta ={10}^{-3}$ (dashed), and $\theta ={10}^{-2}$ (dash-dotted), respectively.