Entanglement as a signature of quantum chaos

We explore the dynamics of entanglement in classically chaotic systems by considering a multiqubit system that behaves collectively as a spin system obeying the dynamics of the quantum kicked top. In the classical limit, the kicked top exhibits both regular and chaotic dynamics depending on the strength of the chaoticity parameter $\kappa$ in the Hamiltonian. We show that the entanglement of the multiqubit system, considered for both the bipartite and the pairwise entanglement, yields a signature of quantum chaos. Whereas bipartite entanglement is enhanced in the chaotic region, pairwise entanglement is suppressed. Furthermore, we define a time-averaged entangling power and show that this entangling power changes markedly as $\kappa$ moves the system from being predominantly regular to being predominantly chaotic, thus sharply identifying the edge of chaos. When this entangling power is averaged over all states, it yields a signature of global chaos. The qualitative behavior of this global entangling power is similar to that of the classical Lyapunov exponent.

Figure 1

The stroboscopic phase-space dynamics of the classical kicked top for $\kappa =3.$ Three hundred stroboscopic trajectories are plotted, each for a duration of 300 kicks.

Figure 2

Dynamical evolution of the linear entropy for initial SCS with $\theta =2.25$ and different $\varphi$. The parameter $\kappa =3$ and the number of qubits is (a) $N=50$, (b) $N=500$.

Figure 3

Dynamical evolution of the concurrence for initial SCS with $\theta =2.25$ and different $\varphi$. The parameter $\kappa =3$ and the number of qubits is $N=50$.

Figure 4

Long-time behavior of the concurrence for the regular case $\left(\theta =2.25,\varphi =0.63\right)$. $\kappa =3$ and $N=50$.

Figure 5

Mean linear entropy and mean concurrence against azimuthal angle $\varphi$. The mean linear entropy is plotted for $N=20$ (solid line), $N=50$ (dashed line), and $N=100$ (dotted line); the mean concurrence is plotted for $N=100$ (solid line), $N=200$ (dashed line), and $N=300$ (dotted line). The parameter $\theta =2.25$ and $\kappa =3$. The time average is over 200 steps, and the subplots below are enlarged versions of above ones.

Figure 6

Contour plot of mean linear entropy against $\varphi$ and $\theta$. The mean is over 200 kicks, $\kappa =3$ and $N=50$.

Figure 7

Contour plot of mean concurrence against $\varphi$ and $\theta$. The average is over 200 kicks, $\kappa =3$ and $N=50$.

Figure 8

Entangling power $e_T$ (square line), $c_T$ (diamond line), and the Lyapunov exponent $\lambda$ (circle line) against $\kappa$. The plot of Lyapunov exponent corresponds to Fig. 1 of Ref. 40. The parameter $N=36$ and time average is over 50 kicks.