Preserving entanglement and nonlocality in solid-state qubits by dynamical decoupling

In this paper, we study how to preserve entanglement and nonlocality under dephasing produced by classical noise with large low-frequency components, such as $1/f$ noise, using dynamical decoupling techniques. We first show that quantifiers of entanglement and nonlocality satisfy a closed relation valid for two independent qubits locally coupled to a generic environment under pure dephasing and starting from a general class of initial states. This result allows us to assess the efficiency of pulse-based dynamical decoupling for protecting nonlocal quantum correlations between two qubits subject to pure-dephasing local random telegraph and $1/f$ noise. We investigate the efficiency of an “entanglement memory” element under two-pulse echo and under sequences of periodic, Carr-Purcell, and Uhrig dynamical decoupling. The Carr-Purcell sequence is shown to outperform the other sequences in preserving entanglement against both random telegraph and $1/f$ noise. For typical $1/f$ flux-noise figures in superconducting nanocircuits, we show that entanglement and its nonlocal features can be efficiently stored up to times one order of magnitude longer than natural entanglement disappearance times employing pulse timings of current experimental reach.

Figure 1

Top panel: Concurrence $C$ as a function of $\gamma t$ under local RTNs with $g>\overline{g}\approx 2.3$ (EWL, initial states with $r=0.91,\left|a\right|=1/\sqrt{2}$): $g=3$ (dashed orange), $g=5$ (thick green), and $g=8$ (dot-dashed purple). We fixed $\delta p_0=0$ except for the thin green line, which corresponds to $\delta p_0=\pm 1$ for $g=5$. In the inset, $g<\overline{g}$: $g=0.5$ (thick black), $g=0.8$ (dashed blue), $g=1.1$ (dot-dashed red), and $g=\overline{g}=2.3$ (dotted magenta). Bottom panel: sketch of the threshold values of the dimensionless coupling parameter $g$ separating dynamical regimes for single qubit and entanglement dynamics.

Figure 2

Concurrence $C\left(2\Delta t\right)$ at the end of the echo as a function of $\gamma t=2\gamma \Delta t$, showing different “entanglement echo” behavior depending on $g\gtrless \overline{g}=2.3$ (for the chosen initial state mentioned in the text). (a) Cases $g_s=g=0.8<\overline{g}$ (dashed black lines) and $g=8>\overline{g}$ (solid red lines), with echo (thick lines) and without (thin lines). For $g=8$, notice the plateau at $\gamma \Delta t=2\pi /g\approx 0.8$. (b) $g_A=0.4$ and $g_B=5$, with echo (thick line) and without (thin line).

Figure 3

Concurrence after an echo sequence $C\left(2\Delta t\right)$, as a function of $g\equiv g_s$ (thick lines). Shown is the behavior for a small pulse interval ($\gamma \Delta t=0.1$ blue dashed line) and for a larger interval ($\gamma \Delta t=1$ orange solid line). For comparison, the concurrence in the absence of echo at the same time $t=2\Delta t$ is reported (thin dashed and solid lines).

Figure 4

Comparison $[lnC\left(\overline{t}\right)]/g^2$, at the fixed time $\gamma \overline{t}=10$ for PDD, as a function of the number of pulses $n$ for RTN (green diamonds) and Gaussian (Ornstein-Uhlenbeck) noise (orange triangles). The values of $g$ are 0.5 (principal panel) and 5 (inset). The initial state is pure maximally entangled.

Figure 5

Concurrence as a function of the dimensionless time $\gamma t$ for $g=0.5$ (top panel) and $g=5$ (bottom panel) for a fixed number of pulses $n=10 \left(\delta p_0=0\right)$. Black dashed lines represent concurrences without DD sequences. The magenta long-dashed line, green dot-dashed line, and orange solid line represent concurrences for PDD, UDD, and CP, respectively. A blue long-dashed line is also displayed coinciding with the magenta one and representing PDD in Gaussian approximation. The gray dotted line at $C_{\mathrm{th}}\approx 0.37$ represents the threshold value of $C$ for Bell inequality violation. The red dotted line at the top is the initial value of concurrence.

Figure 6

Concurrence under CP as a function of the number of pulses $n$ at fixed times $\gamma \overline{t}=\gamma t_{\mathrm{ESD}}$ for $g\le \overline{g}$ and $\gamma \overline{t}=\gamma t_{\mathrm{FD}}$ for $g>\overline{g}$. The values of $g$ are 0.5 (purple circles), $\overline{g}=2.3$ (blue diamonds), and 5 (green squares). The gray dashed line at $C_{\mathrm{th}}\approx 0.37$ represents the threshold value of $C$ after which there is a Bell inequality violation. The red dotted line is the initial value of concurrence.

Figure 7

Concurrence as a function of time $t$ at a fixed number of pulses $n=4$ (top panel) and $n=10$ (bottom panel). The blue long-dashed line, green dot-dashed line, and orange solid line represent concurrences for PDD, UDD, and CP, respectively. The black dashed line is the concurrence in the absence of pulses, with entanglement disappearing at $t_{\mathrm{ESD}}\approx 27\mathrm{ns}$. The gray dotted line at $C_{\mathrm{th}}\approx 0.37$ is the threshold value of $C$ for after Bell inequality violation. The top red dotted line is the initial value of concurrence. The $1/f$-noise figures are given in the text.

Figure 8

Concurrence as a function of the number of pulses $n$ at ${\overline{t}}_1=0.1\mu \mathrm{s}$ (top panel) and ${\overline{t}}_2=0.3\mu \mathrm{s}$ (bottom panel). Blue circles, green squares, and orange diamonds represent concurrences for PDD, UDD, and CP, respectively. The CP points are superposed to the UDD ones. The gray dashed line at $C_{\mathrm{th}}\approx 0.37$ represents the threshold value of $C$ after which there is a Bell inequality violation. The red dotted line at the top is the initial value of concurrence.