Scrambling of quantum information in quantum many-body systems

We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of the integrability of Hamiltonians; TMI can be negative or positive for both integrable and nonintegrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by nonintegrability. Specifically, we argue that there are a few exceptional initial states that do not exhibit scrambling, and show that such exceptional initial states have small effective dimensions. Furthermore, we calculate TMI in the Sachdev-Ye-Kitaev (SYK) model, a fermionic toy model of quantum gravity. We find that disorder does not make scrambling slower but makes it smoother in the SYK model, in contrast to many-body localization in spin chains.

Figure 1

Schematics of our setup. Initially, qubit $A$ is maximally entangled with qubit $B$, while $C$ and $D$ are not correlated with $A$ and $B$. Then, BCD evolves unitarily with a Hamiltonian that is either integrable or nonintegrable, and either clean or disordered. We calculate the real-time dynamics of TMI between $A$, $B$, and $C$.

Figure 2

Time dependence of BMI and TMI for the nonintegrable XXX model with parameters $L=14,J^{\prime }=0.8J$, and $l=1$ or $L/2-1$. The information content and the initial state are (a) BMI, $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$, (b) BMI, all up, (c) TMI, $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$, and (d) TMI, all up.

Figure 3

Dependence on the subsystem size of the decay time $\tau$ of BMI. The system size is $L=128$ and the initial state is all up.

Figure 4

Time dependence of TMI for the integrable XXX model with parameters $L=14,J^{\prime }=0$, and $l=1$ or $L/2-1$. The initial state is (a) $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ and (b) all up.

Figure 5

Initial-state dependence of the maximum (purple) and the minimum (green) values of TMI for the XXX model with parameters $L=12,l=L/2-1$. (a) Nonintegrable case ($J^{\prime }=0.8J$), (b) integrable case ($J^{\prime }=0$).

Figure 6

Effective dimension $D_{\mathrm{eff}}$ vs the minimum value of TMI, $I_3^{\min }$. Parameters are $L=12$ and $l=L/2-1$. (a) XXX model, (b) TFI model.

Figure 7

Initial-state dependence of the maximum (purple) and the minimum (green) values of TMI for the TFI model with parameters $L=12$ and $l=L/2-1$. (a) Nonintegrable case ($h_x=2.1J$ and $h_z=1.1J$), (b) integrable case ($h_x=J$ and $h_z=0$).

Figure 8

Time dependence of BMI and TMI for the disordered XXX model with the initial $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ state with $L=12$. The number of samples is 128. The information content and the phase are (a) BMI, ergodic ($h=J$), (b) BMI, MBL ($h=10J$), (c) TMI, ergodic ($h=J$), and (d) TMI, MBL ($h=10J$).

Figure 9

Time dependence of TMI for the disordered XXX model with the initial all-up state with $L=12$. The number of samples is 128. The phase are (a) ergodic ($h=J$) and (b) MBL ($h=10J$).

Figure 10

Time dependence of TMI for the SYK model with parameters $L=14$, and $l=1$ or $L/2-1$. (a) Disordered SYK model with the initial $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ state. (b) Clean SYK model and typical samples of the disordered SYK model with the initial $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ state. (c) Disordered SYK model with the initial all-up state.

Figure 11

Time dependence of BMI for (a) nonintegrable and (b) integrable XXX models. The system size is $L=128$ and the initial state is all up.

Figure 12

BMI as a function of time $t$ and subsystem size $l$ for (a) nonintegrable and (b) integrable XXX models. The system size is $L=128$ and the initial state is all up.

Figure 13

Time dependence of BMI for the TFI model with $L=14$. The integrability and the initial state are (a) nonintegrable, $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$, (b) nonintegrable, all up, (c) integrable, $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$, and (d) integrable, all up.

Figure 14

Time dependence of TMI for the TFI model with $L=14$. The integrability and the initial state are (a) nonintegrable, $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$, (b) nonintegrable, all up, (c) integrable, $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$, and (d) integrable, all up.

Figure 15

Time dependence of BMI for the disordered XXX model with the initial all-up state with $L=12$. The number of samples is 128. The phase is (a) ergodic ($h=J$) and (b) MBL ($h=10J$).

Figure 16

Time dependence of BMI for the disordered SYK model. The ensemble average is taken over 16 samples. The initial state and the system size is (a) $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$, $L=12$ and (b) all up, $L=10$.