Hayden-Preskill recovery in Hamiltonian systems

Information scrambling refers to the unitary dynamics that quickly spreads and encodes localized quantum information over an entire many-body system and makes the information accessible from any small subsystem. While information scrambling is the key to understanding complex quantum many-body dynamics and is well-understood in random unitary models, it has been hardly explored in Hamiltonian systems. In this Letter, we investigate the information recovery in various time-independent Hamiltonian systems, including chaotic spin chains and Sachdev-Ye-Kitaev models. We show that information recovery is possible in certain, but not all, chaotic models, which highlights the difference between information recovery and quantum chaos based on the energy spectrum or the out-of-time-ordered correlators. We also show that information recovery probes transitions caused by the change of information-theoretic features of the dynamics.

Figure 1

A diagram of the Hayden-Preskill protocol. Time flows from bottom to top. Horizontal lines imply that the qubits connected by the line may be entangled. The initial states on $AR$ and $B{B}^{\prime }$ are given by a maximally entangled state of $k$ ebits, i.e., $k$ EPR pairs, that keeps track of quantum information in $A$, and a purified state $|\xi \left(\beta \right)\rangle$ of a thermal state on $B$ at the inverse temperature $\beta$, respectively. The system $S:=AB$ undergoes Hamiltonian dynamics by ${\widehat{H}}_S$, and then, is split into two subsystems, $C$ of $\ell$ qubits and $D$ of $N-\ell$ qubits. By applying a quantum channel $\mathcal{D}$ onto $B^{\prime }C$, one aims to decode the quantum information in $A$, that is, to recover the $k$ EPR pairs between $\widehat{A}$ and $R$. This protocol has a natural interpretation in the context of information paradox [1]. See also a tutorial [51].

Figure 2

Semilogarithmic plot of the late-time values of $\overline{\mathrm{\Delta }}$ for ${\widehat{H}}_{\mathrm{XXZ}}$, ${\widehat{H}}_{\mathrm{Ising}}$, and ${\widehat{H}}_{{\mathrm{SYK}}_4}$. To compare, the values for the Haar random unitary are also plotted. $k=1$ for all models, and $N=12$ ($N_{\mathrm{q}}=12$) is chosen for the XXZ and Ising $\left({\mathrm{SYK}}_4\right)$ models. The dimension of the Haar random unitary is $2^{12}$. Note that the conservation of the $z$ component of spins (parity) is considered for the Heisenberg spin chain (SYK model). The values of $(g,h)$ are those discussed in [61]. The averages for $t=(1,2,....,10)\times {10}^6$ are plotted as the late-time value. For random-field average, 16 samples are taken.

Figure 3

Semilogarithmic plot of the late-time values of $\underline{\mathrm{\Delta }}$ for ${\widehat{H}}_{\mathrm{XXZ}}$, ${\widehat{H}}_{\mathrm{Ising}}$ ($N=12$), and ${\widehat{H}}_{{\mathrm{SYK}}_4}$ ($N_{\mathrm{q}}=12$). For the Hamiltonians with random field, averages over 16 samples are taken.

Figure 4

Semilogarithmic plot of the value of ${\overline{\mathrm{\Delta }}}_{{\widehat{H}}_{{\mathrm{SYK}}_4}}(t,\beta =0)$ against $t$ for $N_{\mathrm{q}}=13$, and $2\le \ell <N_{\mathrm{q}}-k$. Average over 64 samples is taken. For $\ell =1$, ${\overline{\mathrm{\Delta }}}_{{\widehat{H}}_{{\mathrm{SYK}}_4}}(t,\beta =0)$ is $\sim 1$ for all $t$. The dashed lines represent ${\overline{\mathrm{\Delta }}}_{\mathrm{Haar}}^{\prime }(\beta =0)=2^{\frac{1}{2}(1-\ell )}$ given in Eq. (3) for $\ell =2,3,...,N_{\mathrm{q}}-2$. For smaller values of $N_{\mathrm{q}}$ and finite $\beta$, see S5 [53].

Figure 5

The late-time value of ${\overline{\mathrm{\Delta }}}_{\pm {\mathrm{spSYK}}_4}(t,\beta =0)$ against $\ell$ is plotted for various numbers $K_{\mathrm{cpl}}$ of nonzero coupling constant. We set $N_{\mathrm{q}}$ to 13, and the number of samples is 64. The average for $t=(1,2,...,10)\times {10}^6$ is plotted as the late-time value in all figures.

Figure 6

The late-time value of ${\overline{\mathrm{\Delta }}}_{{\mathrm{SYK}}_{4+2}}(t,\beta =0)$ plotted for various $\ell$ against the value of $\delta$. $N_{\mathrm{q}}$ is set to 13. The number of samplings is 64. The lines connecting the data points are guide to the eye. The horizontal dashed lines indicate ${\overline{\mathrm{\Delta }}}_{\mathrm{Haar}}^{\prime }$ for various $\ell$.