Multipartite entanglement generation and fidelity decay in disordered qubit systems

We investigate multipartite entanglement dynamics in disordered spin-$1∕2$ lattice models exhibiting a transition from integrability to quantum chaos. Borrowing from the generalized entanglement framework, we construct measures for correlations relative to arbitrary local and bilocal spin observables, and show how they naturally signal the crossover between distinct dynamical regimes. Analytical estimates are obtained in the short- and long-time limits. Our results are in qualitative agreement with predictions from the random matrix theory and are relevant to both condensed-matter physics and to the stability of quantum information in disordered quantum computing hardware.

Figure 1

(Color online) Local purity critical time $t_c$ $\mathbf{(}{P}_1\left(t_c\right)=0.9\mathbf{)}$ as a function of $J∕\delta$ for $\delta =0.02,0.05,0.1,0.2$ (diamonds, triangles, circles, squares). To smooth statistical fluctuations, here and in the following we consider averages ${⟨P_1\left(t\right)⟩}_D$ over a number $N_r=10$ of disorder realizations. We assume $n=10$ unless otherwise specified. Inset: Time evolution of $P_1\left(t\right)$ in the FGR $J=\delta ∕10=0.01$ (red) and ergodic $J=\delta =0.1$ (blue) regime starting from the state of Eq. (5) for $n=14$, $n_B=0$. Dotted lines: exponential and Gaussian fits, as in Eq. (3).

Figure 2

(Color online) Saturation values of $I\text{−}1$ (empty symbols) and of $(P_1^{-1}-1)R$ (filled symbols) vs $J∕\delta$ for $\delta =0.05$ (circles), $\delta =0.1$ (squares), $\delta =0.2$ (diamonds), and the rescaling factor $R=7$. The straight line is proportional to $C{(J∕\delta )}^2{N}_B(k=n∕2)$, $C\sim 0.7$. Inset: Saturation value of the IPR (triangles up) and $P_1^{-1}$ (triangles down) in the ergodic regime $(\delta =0,J=0.1)$ as a function of $n$. The dashed line gives $N_B(n∕2)$ vs $n$.

Figure 3

(Color online) Local purity $P_1\left(t\right)$ vs time in the FGR (black lines) and the ergodic regime (red lines) for different initial states with $n_B=0,1,2,3$, $\delta =0.1$, and $J=0.01$ (black curves), $J=0.1$ (red curves). Dashed lines: Different separable initial states $\mid {\psi }_0⟩=\mid 0101010110⟩,\mid 010101010⟩\otimes (\mid 0⟩+\mid 1⟩)∕\sqrt{2}$ for $\delta =0.1,J=0.01$. Inset: The sames curves rescaled by $P_1\left(0\right)$. Also shown (dot-dashed lines) is the local purity decay in the ergodic and the FGR regimes for an initial $\mid W⟩$ state.

Figure 4

(Color online) Purity critical time $t_c$ as a function of $J∕\delta$ for different initial states, $n_B=1$ (circles), $n_B=2$ (squares), $n_B=3$ (diamonds), $\mid W⟩$ (blue triangles), and different purity measures: $P_1\left(t\right)$ (empty) and $P_2\left(t\right)$ (full). Inset: $P_2\left(t\right)$ vs time in the FGR regime (black lines) and in the ergodic regime (red lines) for different initial states with $n_B=0,1,2,3$.