Speed of quantum information spreading in chaotic systems

We present a general theory of quantum information propagation in chaotic quantum many-body systems. The generic expectation in such systems is that quantum information does not propagate in localized form; instead, it tends to spread out and scramble into a form that is inaccessible to local measurements. To characterize this spreading, we define an information speed via a quench-type experiment and derive a general formula for it as a function of the entanglement density of the initial state. As the entanglement density varies from zero to one, the information speed varies from the entanglement speed to the butterfly speed. We verify that the formula holds both for a quantum chaotic spin chain and in field theories with an AdS/CFT gravity dual. For the second case, we study in detail the dynamics of entanglement in two-sided Vaidya-AdS-Reissner-Nordstrom black branes. We also show that, with an appropriate decoding process, quantum information can be construed as moving at the information speed, and, in the case of AdS/CFT, we show that a locally detectable signal propagates at the information speed in a spatially local variant of the traversable wormhole setup.

Figure 1

Panel A (top): Conventional Hayden-Preskill protocol for a spatially local scrambler. The right side of the dot-dashed line is the memory which is maximally entangled with the system at time zero and frozen thereafter. The left side of the dot-dashed line contains the system which undergoes local scrambling unitary dynamics. Once the information correlated with the reference (red circle) spreads over the whole system, access to the memory plus a few bits of the system suffice to recover the entanglement with the reference. Panel B (bottom): For $f=0$ (or more generally $f<1$), there is no (or submaximal) entanglement with the memory. Instead, the local scrambling dynamics will also dynamically generate the memory in addition to spreading the information. The bottleneck step turns out to the generation of the memory, hence relating the success of this generalized Hayden-Preskill protocol to the generation of maximal entanglement.

Figure 2

(a) Comparing the information expansion between a product state and a maximally entangled state. Evidently, the information expansion in the maximally entangled case is faster. (b) The information speeds increases with the entanglement density $f$, up to approximately the butterfly speed as $f$ goes to 1 and is tracked by $v_E\left(f\right)/(1-f)$. The entanglement speed as a function $f$ is shown in the inset.

Figure 3

Penrose diagram of the planar, 1-sided AdS-RN-Vaidya spacetime. The infalling shell is in blue. The infalling particle is in green. The horizons are in dashed gray and the timelike singularities in red.

Figure 4

Phase diagram with the parameters $z_{+}=1$ and $z_{-}=1.001$. The blue dots lying on the green curve are numerically computed, and the green curve is the interpolation of the blue dots.

Figure 5

Left panel: Red and green dots are $v_I$ and $v_E\left(f\right)/(1-f)$, respectively, computed numerically for a range of values of the entanglement fraction $f$. The blue line is $v_E\left(f\right)/(1-f)$ as obtained through equation (29). Right panel: Green dots show numerically obtained values of $v_E$, while the blue line is $v_E\left(f\right)$ from equation (29).

Figure 6

$v_I$ as a function of $f$ (blue dots). The horizontal blue line is $v_E$, and the horizontal orange line is $v_B$.

Figure 7

Setup of the traversable wormhole calculation. The coupling between the two sides gives rise to a wave function which has negative null energy (in red). The message (in green) undergoes a negative Shapiro delay.

Figure 8

Plot of the imaginary part of $C$ as a function of $T$ and $X$, for two choices of parameters. Both plots use ${\mathrm{\Delta }}_{\mathcal{O}}={\mathrm{\Delta }}_{\psi }=0.7$, and $t_{\mathrm{scr}}=5$. The upper figure has $g=30$ while the lower figure has $g=300$.

Figure 9

Upper left: Plot of $z\left(x\right)$. Upper right: Plot of $z\left(x\right)$ zoomed into the coordinate range $z_{f,+}\le z\le z_m$. Lower left: Plot of $z\left(v\right)$. Lower right: Plot of $z\left(v\right)$ zoomed into the coordinate range $z_{f,+}\le z\le z_m$. We only plotted over positive values of $x$ for both the upper left and lower left panels; the plot for negative values of $x$ can be obtained by reflection symmetry about $x=0$.

Figure 10

Plot of the area of the HRT surface versus boundary time, with the half-width kept fixed at the same value as in the previous plot. The dashed red vertical line is the saturation time.

Figure 11

Left panel: Phase diagram for the case $f=0.5$. Right panel: Phase diagram for the case $f=0.99$.

Figure 12

Schematic illustrating the decomposition of the time-evolution operator, $U\approx U_{A^{\prime }}{U}_{A^c}$, along the effective light-cone set by the butterfly velocity $v_B$ (or by the light-cone speed $v_L$, when this is not equal to $v_B$).