Realizing the Hayden-Preskill protocol with coupled Dicke models

Hayden and Preskill proposed a thought experiment in which Bob can recover the information Alice throws into a black hole if he has a quantum computer entangled with the black hole, and for which Yoshida and Kitaev recently proposed a concrete decoding scheme. In the context of quantum many-body physics, the parallel question is that after a small system is thermalized with a large system, how can one decode the initial state information with the help of two entangled many-body systems? Here, we propose to realize this decoding protocol in a physical system of two Dicke models, with two cavity fields prepared in a thermofield double state. We show that the Yoshida-Kitaev protocol allows us to read out the initial spin information after it is scrambled into the cavity. We show that the readout efficiency reaches a maximum when the model parameters are tuned to the regime where the system is the most chaotic, characterized by the shortest scrambling time in the out-of-time-ordered correlation function. Our proposal opens up the possibility of discussing this profound thought experiment in a realistic setting.

Figure 1

(a) Schematic of our physical system of a coupled Dicke model, where TFD stands for the thermofield double state. (b) Diagrammatical illustration of the Yoshida and Kitaev version of the Hayden-Preskill protocol.

Figure 2

Thermalization and OTOC for a single Dicke model. (a) The probability of projection onto the four Bell states formed by $\mathcal{A}$ and $\mathcal{R}$. (b) OTOC for the Dicke model with three different coupling constants $g/\hslash {\omega }_0$. Both (a) and (b) are plotted in terms of $t$ (in units of $1/{\omega }_0$). Here, $\beta =0.01/\hslash {\omega }_0$ and $\hslash {\omega }_z=3\hslash {\omega }_0$.

Figure 3

(a) The conditional probability $\mathcal{F}$ as a function of $t$ for three different coupling constants $g$. (b) $\mathcal{F}$ and an OTOC between spin and an operator ${\widehat{\mathcal{O}}}_{\mathcal{D}}$ in Hilbert space $\mathcal{D}$ are plotted in the same figure with the same coupling constant $g=2$.

Figure 4

The conditional probability $\mathcal{F}$ for (a) $\delta g$ and (b) $\delta {\omega }_0$, where $\delta g$ and $\delta {\omega }_0$ is the difference in coupling constant and difference in cavity photon frequency, respectively. We also plot the correlation function $\langle {\widehat{\mathcal{O}}}_{\mathcal{D}}\widehat{\mathcal{S}}{\widehat{\mathcal{O}}}_{\mathcal{D}}^{†}{\widehat{\mathcal{S}}}^{†}\rangle$ that is closely related to this dependence.