Entanglement across a transition to quantum chaos

We study the relation between entanglement and quantum chaos in one- and two-dimensional spin-$1∕2$ lattice models, which exhibit mixing of the noninteracting eigenfunctions and transition from integrability to quantum chaos. Contrary to what occurs in a quantum phase transition, the onset of quantum chaos is not a property of the ground state but takes place for any typical many-spin quantum state. We study bipartite and pairwise entanglement measures—namely, the reduced von Neumann entropy and the concurrence—and discuss quantum entanglement sharing. Our results suggest that the behavior of the entanglement is related to the mixing of the eigenfunctions rather than to the transition to chaos.

Figure 1

Level statistics parameter $\gamma$ as a function of the coupling parameter $J$ (in units of $\delta ∕L$), for the model of Eq. (18) for a $3\times 3$ lattice, calculated from the energy levels in the spectral band centered at $-{\Delta }_0$ averaged over 2000 random realizations of ${\delta }_i$ and $J_{i,j}$. The other parameters are ${\Delta }_0=1$ and $\delta =0.09$.

Figure 2

Participation number $\xi$ as a function of $JL∕\delta$ for the model of Eq. (18) for a $3\times 3$ lattice, with parameter values as in Fig. 1. The PN was obtained from the central eigenfunction of the band centered at $-{\Delta }_0$ averaged over 2000 random realizations. The dashed line corresponds to $N_b∕3$.

Figure 3

Concurrence $C_n$ as a function of the coupling parameter $JL∕\delta$ for the model of Eq. (18) for a $3\times 3$ lattice, with parameter values as in Fig. 1. $C_n$ was obtained from the central eigenfunction of the central negative band of the spectrum and averaged over all possible pairs of (a) nearest-neighbor qubits $(n=1$, squares) and (b) next-nearest-neighbors qubits $(n=2$, circles).

Figure 4

(Color online) Density map for the concurrence $C_1$ calculated for each individual eigenfunction (index in the vertical axis), averaged over 200 random realizations. $C_1$ is plotted as a function of the coupling parameter $JL∕\delta$ for the model of Eq. (18) for a $3\times 3$ lattice, with parameter values as in Fig. 1. In the inset, we show $C_1$ averaged over the central $1∕3$ of the eigenstates of each individual band. The different curves correspond to the bands for states with 4 (solid line), 3 (dotted line), 2 (dashed line), 1 (long-dashed line), and 0 (dot-dashed line) spins “up.”

Figure 5

Nearest-neighbor level spacing distribution $P\left(s\right)$ for the AA model $(l_c=L-1)$ of Eq. (25) for a chain of $12\text{qubits}$ and (a) $J∕J_c=0.35$ and (b) $J∕J_c=15$, calculated from the energy levels in the central band of the spectrum and averaged over ten random realizations. The solid lines correspond to the Poisson (a) and Wigner surmise (b) theoretical distributions.

Figure 6

Participation number $\xi$ as a function of the coupling parameter $J$ for the AA model of Eq. (25), for a chain of size $L=12$. The PN is averaged over all eigenfunctions in the central band of the spectrum and over ten random realizations. The dashed horizontal lines correspond to $N_b∕3$ and $N∕3$ where $N=2^L$ is the dimension of the Hilbert space.

Figure 7

Concurrence as a function of the coupling parameter $J$ for the model of Eq. (25) with interaction range $l_c=5$ for a chain of ten qubits. For each eigenfunction $C_n$ was obtained as the average concurrence between all possible pairs of qubits separated by a distance $n$: $n=1$ for nearest-neighbors pairs (diamonds), $n=2$ for next-nearest-neighbors pairs (right triangles), $n=3$ (circles), $n=4$ (squares), and $n=5$ (up triangles). The plotted values correspond to $C_n$ averaged over all the eigenstates of the central band and over 30 random realizations.

Figure 8

Mean concurrence $C_3$ between qubits at a distance $n=3$, averaged over all the eigenfunctions in the central band of the spectrum and over 30 random realizations. The mean concurrence is plotted as a function of the coupling parameter $J$ for the model of Eq. (25) in a chain of ten qubits. We compare $C_3$ for different range of the interaction: The NN model corresponding to $l_c=1$ (diamonds), $l_c=2$ (triangles), and $l_c=3$ (circles).

Figure 9

Mean concurrence averaged over all possible pairs of qubits for all the eigenstates in the central band of the spectrum and over 30 random realizations for the AA model of Eq. (25) $(l_c=L-1)$ as a function of the coupling parameter $J$ and for different sizes of the chain of qubits: $L=6$ (diamonds), $L=8$ (triangles), $L=10$ (circles), and $L=12$ (squares).

Figure 10

Normalized distribution $P\left(c_{\lambda }\right)$ of the quantity $c_{\lambda }={\lambda }_1-{\lambda }_2-{\lambda }_3-{\lambda }_4$ of Eq. (5) for the AA model, Eq. (25), with $L=12$ and coupling: (a) $J∕J_c=0.01$ and (b) $J∕J_c=100.0$. In the inset the behavior of the first moment of the distribution $⟨c_{\lambda }⟩$ is shown as a function of the coupling parameter $J$.

Figure 11

von Neumann entropy $S_1$ between one qubit and the rest of the system; $S_1$ is averaged over all qubits and over all eigenstates in the central band of the spectrum and over 30 random realizations for the AA model of Eq. (25) as a function of the coupling parameter $J$ and for different sizes of the chain of qubits: $L=6$ (diamonds), $L=8$ (triangles), $L=10$ (circles), and $L=12$ (squares).

Figure 12

von Neumann entropy $S_n$ between left and right blocks of sizes $n$ and $L-n$ qubits, respectively, averaged over all eigenfunctions in the central band of the spectrum and over ten random realizations. The data correspond to the AA model of Eq. (25) for a chain of size $L=12$. From bottom to top, the different curves are from $n=1$ to $n=5$. In the inset $S_n(J∕J_c=15)$ is plotted as a function of $n$. The dashed line corresponds to a linear fit with a slope of $0.93\pm 0.02$.

Figure 13

von Neumann entropy $S_{L∕2}$ as a function of the strength of the interaction $J$, averaged over all eigenfunctions in the central band of the spectrum and over ten random realizations. The data correspond to the AA model of Eq. (25) for a chain of sizes $L=6$ (diamonds), $L=8$ (triangles), $L=10$ (circles), and $L=12$ (squares). In the inset $S_{L∕2}(J∕J_c=15)$ is plotted as a function of $L$. The dashed line corresponds to a linear fit with a slope of $0.52\pm 0.02$.

Figure 14

Comparison between the NN model Eq. (22) (open circles) and the model of Eq. (25) with $l_c=L∕2$ (solid squares), for a chain of qubits of length $L=10$. From top to bottom, the three panels show, as a function of the coupling parameter $J$; the behavior of the spectral statistics parameter $\gamma$ (top panel), the mean concurrence $C_1$ between nearest-neighbor pairs of qubits (middle panel), and the participation number $\xi$ (bottom panel). All quantities were calculated in the central band of the spectrum and averaged over 30 random realizations.