Dynamical entanglement percolation with spatially correlated disorder

TL;DR

This work studies dynamical entanglement percolation in quantum networks where edge-specific two-qubit interactions generate time-dependent entanglement, analyzed through percolation theory. Edge states yield Schmidt coefficients λ_e(t) = rac{1 + | ext{cos}( ext{ω}_e t)|}{2} and singlet-conversion probabilities φ_e(t) = 1 - | ext{cos}( ext{ω}_e t)|, giving time-dependent active-edge fractions p(t) and giant-component fractions P(t). For uncorrelated disorder, P(p) aligns with the standard bond-percolation curve and a universal asymptotic p_ obreak extunderscore ∞ = 1 - rac{2}{ ext{π}}; however, spatial correlations drive hysteresis and multi-valued P(p), motivating a two-colour bond-percolation description with mean-field support via a critical line 3 φ_1 φ_2 - φ_1 - φ_2 + 1 = 0. The results reveal richer percolation phenomena in dynamical quantum networks than in static models, informing design principles for robust long-range entanglement under realistic correlated disorder.

Abstract

The distribution of entanglement between the nodes of a quantum network plays a fundamental role in quantum information applications. In this work, we investigate the dynamics of a network of qubits where each edge corresponds to an independent two-qubit interaction. By applying tools from percolation theory, we study how entanglement dynamically spreads across the network. We show that the interplay between unitary evolution and spatially correlated disorder leads to a non-standard percolation phenomenology, significantly richer than uniform bond percolation and featuring hysteresis. A two-colour correlated bond percolation model, whose phase diagram is determined via numerical simulations and a mean-field theory, fully elucidates the physics behind this phenomenon.

Figures (10)

• Figure 1: Schematic representation of a quantum network. We study a quantum network made of stations, forming the nodes of a graph, connected by edges that correspond to entangling interactions between pairs of qubits. Big circles denote stations, small circles represent qubits, lines indicate the presence of an interaction between two qubits. The interaction $H_e$ between the qubits $i_X=3$ and $j_Y=2$, represented by the edge $e=\varepsilon_{32}^{XY}$, is highlighted in red. Note that each interaction involves two dedicated qubits, so that the number of qubits belonging to each station is equal to the degree of the corresponding node of the graph.
• Figure 2: Schematic representation of dynamical entanglement percolation . (a) The dashed lines represent quantum states evolving as in Eq. \ref{['eq:time_evolution_single_edge']}. In this example, the network is a distortion of a square lattice, and spatial correlations are present between neighboring edges. (b) Result of the singlet conversion (SC) process. At a certain time $t$, each edge state $\ket{\psi_e(t)}$ is converted to a maximally entangled state with the singlet conversion probability $\phi_{e}(t)$ given in Eq. \ref{['eq:activation_probability']}. Successful conversions are represented with continuous lines. The maximally entangled connected components are highlighted in blue, with thicker lines for the largest component (a proxy for giant component). The probability that $X$ and $Y$ can share entanglement is related to the size of the giant component. In this case, they both belong to the giant component.
• Figure 3: Dynamical entanglement percolation observables on two-dimensional square lattice with i.i.d. frequencies. (a)-(d) The observables $p(t)$ (solid lines) and $P(t)$ (symbols and dashed lines) for independent Gaussian frequencies with mean $\Omega=1$ and (a) $\sigma_{\omega}=0$ (uniform frequencies), (b) $\sigma_{\omega}=0.1$, (c) $\sigma_{\omega}=0.2$, (d) $\sigma_{\omega}=0.3$. For uniform frequencies in panel (a), both $p$ and $P$ are periodic with period $T=\pi$. In all other cases, $p(t) \to p_\infty=1-2/\pi$ for large times. (e) Parametric plot of $P(p)$ using the observables $p(t)$ and $P(t)$ in panels (a)-(d). The effect of the dynamical evolution disappears when observing the order parameter $P(p)$, as all curves are equivalent and they all reproduce uniform bond percolation $P_0(p)$ (red line). The role of $t$ is just manifest in the fact that, depending on $\sigma_{\omega}$, not all values of $P$ can be reached. The inset in panel (e) shows the behavior of $P(p)$ close to the bond percolation threshold $p_c=1/2$. All data perfectly overlap. For all plots, the system size is $L=10^3$, hence $N=10^6$. All results are averaged over $10$ realizations of the disorder, and over $100$ realizations of the edge activation process.
• Figure 4: Dynamical entanglement percolation observables on perturbed square lattice with exponentially decaying frequencies. Results of numerical simulations on a perturbed square lattice with noise strength $\sigma$ and frequencies depending on the distances as $\omega(d)=\Omega e^{-d/\lambda}$. We set $\Omega=2$, $\lambda=2$. (a) The observables $p(t)$ (solid lines) and $P(t)$ (symbols) for $\sigma=0.1$, and (b) the corresponding parametric plot of $P(p)$. (c)-(d) As in (a)-(b) but for $\sigma=0.2$. In both cases, the dynamics converges to the asymptotic value $p_\infty=1-2/\pi$, after a preasymptotic oscillatory behavior which depends on the noise amplitude. (e) The behavior of the observables $P(p)$ as in panel (b) and (d) close to the uniform percolation threshold on the square lattice $p_c=1/2$stauffer2018introduction. In contrast to the case of independent random frequencies, $P$ clearly depend on $\sigma$, and the order parameter $P(p)$ is no longer the same as uniform bond percolation.
• Figure 5: Dynamical entanglement percolation observables on perturbed square lattice with correlated Bernoulli frequencies. Results of numerical simulations on a perturbed square lattice with noise strength $\sigma$ and frequencies depending on the distances as in Eq. \ref{['eq:frequencies_threshold']}. (a) The observables $p(t)$ (solid lines) and $P_C(t)$ (symbols and dashed lines) in the correlated case for $\sigma=0.1$, $\widetilde{\Omega}=2$, $\lambda=1$, and (b) the corresponding parametric plot of $P_C(p)$. (c)-(d) As in (a)-(b) but for the uncorrelated case. (e) The behavior of the observables $P_C(p)$ and $P_U(p)$ as in panel (b) and (d), respectively, close to the uniform percolation threshold on the square lattice $p_c=1/2$stauffer2018introduction. (f)-(j) as in (a)-(e) but for $\sigma=0.2$, $\widetilde{\Omega}=5/2$, $\lambda=1$. The comparison between the correlated and the uncorrelated scenario shows that the order parameters $P_C$ clearly depend on the parameters $\sigma$ and $\widetilde{\Omega}$, exhibiting a nontrivial behavior with coexisting branches.