Entangling power of time-evolution operators in integrable and nonintegrable many-body systems

The entangling power and operator entanglement entropy are state-independent measures of entanglement. Their growth and saturation is examined in the time-evolution operator of quantum many-body systems that can range from the integrable to the fully chaotic. An analytically solvable integrable model of the kicked transverse-field Ising chain is shown to have ballistic growth of operator von Neumann entanglement entropy and exponentially fast saturation of the linear entropy with time. Surprisingly, a fully chaotic model with longitudinal fields turned on shares the same growth phase, and is consistent with a random matrix model that is also exactly solvable for the linear entropy entanglements. However, an examination of the entangling power shows that its largest value is significantly less than the nearly maximal value attained by the nonintegrable one. The importance of long-range spectral correlations, and not just the nearest-neighbor spacing, is pointed out in determining the growth of entanglement in nonintegrable systems. Finally, an interesting case that displays some features peculiar to both integrable and nonintegrable systems is briefly discussed.

Figure 1

Operator von Neumann (top) and linear (bottom) entanglement entropies, $E_{vN,l}\left(U^n(\tau ,h)\right)$ for nonintegrable (Set NI) and integrable cases (Set I), with $\tau =\pi /4$ and $L=10$ spins. Also shown are the corresponding RMT model results. Inset shows the ratio of the entropies for the nonintegrable model with the corresponding RMT averages from Eqs. (3).

Figure 2

Schematic of the labeling scheme used for the two partitions $A$ and $B$.

Figure 3

The von Neumann (top) and linear (bottom) entangling powers, ${\text{ep}}_{vN,l}\left(U^n(\tau ,h)\right)$ for the nonintegrable (Set-NI) and the integrable cases (Set-I), with $\tau =\pi /4$ and $L=10$ spins. Also shown are the corresponding RMT model results. Inset shows the ratio of the entangling powers for the nonintegrable model with the corresponding RMT averages in Eqs. (3).

Figure 4

Linear entropy operator entanglements $E_l\left(U^n(\tau ,h)\right)$ and $E_l\left(U^n(\tau ,h)S\right)$ for integrable (Set-I, top) and nonintegrable cases (Set-NI, bottom), with $\tau =\pi /4$ and $L=10$ spins. Inset shows details of $E_l\left(U^n(\tau ,h)S\right)$ for the nonintegrable case and also the RMT model prediction (triangles).

Figure 5

The von Neumann entropy entangling power, ${\text{ep}}_{vN}\left(U^n(\tau ,h)\right)$ for the nonintegrable Set-NI, integrable Set-I, with $\tau =\pi /8$ and $L=10$ spins. Also shown are the corresponding RMT model results.

Figure 6

Distribution of spacing for the nonintegrable Set-NI, with $\tau =\pi /3,\pi /4,\pi /8$ for $L=14$. Inset shows the mean of the ratio of nearest-neighbor spacing distribution (NSS) of the operator $U$ vs time between kicks $\tau$. Dots in the inset correspond to $\tau =\pi /3,\pi /4$ and $\pi /8$.

Figure 7

The von Neumann entropy entangling power, ${\text{ep}}_{vN}\left(U^n(\tau ,h)\right)$ for the nonintegrable Set-NI, integrable Set-I, with $\tau =\pi /3$ and $L=10$ spins. Also shown are the corresponding RMT model results.

Figure 8

Number variance for $\tau =\pi /3,\tau =\pi /4$, and $\tau =\pi /8$. Inset shows the same for $\tau =\pi /3,\pi /4$ in a longer range.

Figure 9

Operator von Neumann entropy (top) $E_{vN}\left(U^n(\tau ,h)\right)$ and entangling power (bottom) when $h_x^i=1.0,h_y^i=0,h_z^i=1$, and $\tau =\pi /4$. The nonintegrable case Set-NI is shown in the latter figure for comparison.

Figure 10

Ancilla picture for the spin chain: the spins in the lower row are subjected to the spin chain dynamics while the upper (ancilla) are not. Initially, corresponding pairs of spins between the chain and the ancilla are maximally entangled in a Bell state $|{\mathrm{\Phi }}^{+}\rangle .$ To find the operator entanglement of $U^{n}S$, one needs to find the entanglement across the $A_1{A}_1^{\prime }..A_{L/2}{A}_{L/2}^{\prime }$ and $B_1{B}_1^{\prime }..B_{L/2}{B}_{L/2}^{\prime }$ bipartition of the state obtained by the action of $U^{n}S$ on the $A_1{A}_2..A_{L/2}{B}_1{B}_2..B_{L/2}$ subsystem.