Signatures of chaos in the dynamics of quantum discord

We identify signatures of chaos in the dynamics of discord in a multiqubit system collectively modelled as a quantum kicked top. The evolution of discord between any two qubits is quasiperiodic in regular regions, while in chaotic regions the quasiperiodicity is lost. As the initial wave function is varied from the regular regions to the chaotic sea, a contour plot of the time-averaged discord remarkably reproduces the structures of the classical stroboscopic map. We also find surprisingly opposite behavior of two-qubit discord versus entanglement of the two qubits as measured by the concurrence. Our results provide evidence of signatures of chaos in dynamically generated discord.

Figure 1

Classical stroboscopic map for the kicked top. The direction of the angular-momentum vector is plotted after each kick for different initial conditions with $p=\pi /2,\kappa =3$.

Figure 2

Dynamical evolution of discord for initial SCS with $\theta =2.25$ and different $\varphi$. Here we have taken parameters as $\kappa =3$ and $j=40$.

Figure 3

Dynamically generated discord as a function of the number of applications of the Floquet map for $j=4$ and $\kappa =3$. (a) Fixed-point initial condition $\left(\theta =2.25,\varphi =0.63\right)$. (b) Regular initial condition $\left(\theta =2.25,\varphi =0.90\right)$. (c) Edge-of-chaos initial conditions $\left(\theta =2.25,\varphi =1.05\right)$. (d) Chaotic initial condition $\left(\theta =2.25,\varphi =2.00\right)$.

Figure 4

Side-by-side comparison showing dynamically generated discord as a remarkable signature of classical chaos in a mixed phase space $(p=\pi /2,\kappa =3$, and $j=40)$. (a) Classical phase space, Poincaré section. (b) Long-time average discord, $D$, as a function of mean coordinate of the initial projected coherent state. A weighted average of $D$ according to the measure on phase space gives the value of $\overline{D}=0.275$ in the chaotic sea and $\overline{D}=0.143$ in the regular islands.

Figure 5

Average discord, $D$, as a function of the chaoticity parameter, $\kappa$, for the kicked top for $j=100$. The average is calculated over the first 350 kicks. The initial state is the same as in Fig. 3, given by $\left(\theta =2.25,\varphi =1.05\right)$.

Figure 6

(a) Contour plot of discord for $j=40$. Blue regions indicate regions of low average discord. (b) Average discord for $\theta =2.25$ and varying $\varphi$. Comparison to classical phase space shows that chaotic region initial conditions lead to higher average discord.

Figure 7

(a) Discord dynamics. (b) Entropy dynamics and (c) concurrence dynamics for $j=4$.

Figure 8

Dynamically generated discord as a function of the number of applications of (a) the Floquet map corresponding to global chaos for $j=4$. (b) A random unitary chosen from the COE for $j=4$. (c) The Floquet map corresponding to global chaos for $j=40$. (d) A random unitary chosen from the COE for $j=40$.

Figure 9

Fluctuations around the mean value of quantum discord in the fully chaotic phase space, $p=\pi /2,\kappa =6$, as measured by the standard deviation as a function of the number of qubits $N$ (blue curve). The red line shows the $1/\sqrt{N}$ behavior.