Quantum teleportation between simulated binary black holes

Abstract

The quantum description of a black hole predicts that quantum information hidden behind the event horizon can be teleported outside almost instantaneously. In this work, we demonstrate that a chiral spin-chain model, which naturally simulates a binary black hole system, can realise this teleportation process. Our system captures two essential components of this protocol: Hawking radiation, which generates the necessary entanglement between the black holes, and optimal scrambling, which enables high-fidelity teleportation on short timescales. Through numerical simulations, we quantify the key timescales governing the process, including the Page time, radiation time, scrambling time, and butterfly velocity, showing their universal dependence on the chiral coupling strength. Our results establish the feasibility of simulating quantum properties of black holes within condensed matter systems, offering an experimentally accessible platform for probing otherwise inaccessible high-energy phenomena.

Figures (11)

• Figure 1: (a) The spin-$1/2$ chiral spin-chain \ref{['ham']}, acts as a quantum simulator of black holes. We take constant coupling $u = 1$ and a spatially varying $v(x) = \alpha \bigl(1 - \tanh[\beta(x - x_h)]\bigr)$. This profile creates two distinct regions: a strongly interacting chiral region ($\langle \chi \rangle \neq 0$) for $0 < x < x_h$ where $v > 2u$, and a weakly interacting non-chiral region ($\langle \chi \rangle = 0$) for $x > x_h$ where $v < 2u$. In the weak-fluctuation regime ($v \lesssim 2u$), mean-field theory applies and the continuum limit corresponds to Dirac fermions propagating on a curved black hole background described by metric \ref{['eqn:metric']}. In this regime the chaotic behaviour is weak. The horizon lies at $x = x_h$, defined by $v = 2u$. Inside the black hole ($v > 2u$), the MFT description fails and strong interactions give rise to optimal scrambling, as expected in the black hole interior. (b) Schematic of a quantum-field description of a black hole in the Gullstrand-Painlevé coordinates. Increasing curvature progressively tilts the lightcones; beyond the horizon, both lightcone directions point inward, preventing classical particles to escape. Deep inside the black hole, large curvature ultimately invalidates the semiclassical geometric picture, leading to strong quantum fluctuations and rapid scrambling of information.
• Figure 2: (a) Schematic of the chiral spin-chain with the encoded state $\ket{\psi}_A$ and the initially prepared EPR states, $\ket{epr}$. To implement the Hayden-Preskill protocol we employ the scrambling Hamiltonian $H$ inside the black hole and $-H$ on its outside, with left and right halves of the chain corresponding geometrically to mirrored black holes. The qubits Alice and Bob initially have access to are labelled A and B respectively. (b) The coupling profile for $u$ (red) and $v$ (green/purple) along the spin-chain for the Hamiltonian in Eq. \ref{['ham']}. Using Eq. \ref{['eqn:couplings']} and taking $\alpha=v$, we highlight the difference in the $v$ coupling between the (smooth purple) geometric interpretation which takes varying $\beta$ where large free system can be numerically considered and the (dashed green) teleportation protocol in a small strongly interacting system where $\beta=\infty$. To realise the Hamiltonian $H\oplus (-H)$ the two-site $u$ term flips after the site $(N-1)/2$, while the 3-site $v$ term is 0 on site $(N-1)/2$ but becomes $\pm v$ on either side.
• Figure 3: (a) Fidelity of teleportation over time, $t$, scaled by $v$ after quenching the initial state, $\ket{\psi}_A\otimes\ket{EPR}$, with Hamiltonian \ref{['ham']}. Here we take $\alpha=v$ and $\beta=\infty$ in Eq. \ref{['eqn:couplings']} with the slight modification such that no $v$ coupling is allowed across the two halves of the chain as shown in Fig. \ref{['fig:sketch']}(b). Success is determined by the overlap between the final state $\ket{\psi}_B$ of the last site with the initially prepared state $\ket{\psi}_A$. Colour indicates the value of $v$ during the quench where we see a notable change in teleportation success across the phase transition at $v/u\approx 2$. Four EPR pairs, $\ket{epr}$, are measured in this case. (b) Similarly but unscaled in time and at a fixed $v=8$. Colour corresponds to a different number of EPR pairs measured, $E$, where we see teleportation success increases with $E$ as expected. Here we take $N=17$ and $u=1$ using exact diagonalisation (ED). (c) Taking a slice at $t=30$ in (b) for different system sizes, we see how the scaling of fidelity with $E$ changes for system size $N$ when compared to the theoretical upper bound of $F_E$. We see with increasing system size our results tend towards the optimal theoretical prediction. Code used is publicly available at danielgithub.
• Figure 4: Page curve, Page time, and Hawking temperature of the mean-field (MF) chiral Hamiltonian \ref{['eq:MFT']}. (a) The entanglement entropy, $S_E$ (blue line), of the black hole across the horizon as a function of time, $t$, during the Hawking radiation of a single particle. This curve, known as the Page curve, reaches its maximum at the Page time, $t_{\rm{Page}}$, when the population inside the horizon (red line) is halved. To accelerate black hole evaporation, we consider a single-particle population inside the black hole and place the horizon at $n_h=2$, while the entropy is evaluated for a bipartition at $n=22$ to account for the entropy of particles that are completely free from the black hole attraction. The inset sketches illustrate the evolution of an EPR pair as it crosses the event horizon (dashed line): at early times, the EPR pair is entirely inside the black hole ($S_E = 0$); at $t_{\rm{Page}}$, half of the pair escapes, yielding maximal entanglement ($S = \ln 2$); at late times, the second half is also emitted, leading to $S \to 0$. (b) The Hawking temperature $T_H$ as a function of $\alpha$ in the MF model (blue line) compared to its analytical value $T_H = \alpha \beta / 2 \pi$ (red line). Here, $N = 500$ and the horizon is at $n_h = 250$. (c) Page time $t_{\rm{Page}}$ as a function of $T_H$ in the MF model (black dots), alongside the fitted curve $A/T_H$ (blue line) with $A = 0.278$, obtained numerically by varying $\alpha$ and $\beta$.
• Figure 5: The Lyapunov exponent $\lambda$ of the chiral spin-chain model \ref{['ham']} in the strongly interacting regime where $v=8$, $u=1$. For small temperatures $T$ the Lyapunov exponent takes the form $\lambda = aT+c$ for $a$ and $c$ some fitting parameters. As $N$ increases the fitting parameters approach $a \to 2\pi \frac{v}{2}$ and $c \to 0$, thus recovering the optimal scrambling behaviour in the thermodynamic limit. Black dashed lines are for guiding the eye. Results are computed using exact diagonalisation (ED) for smaller system sizes, and a Krylov subspace method (see Methods \ref{['sec:krylov']} for details) to reach larger system sizes. We see excellent agreement for the two when ED is possible. ED results are from $N=6$ to $N=13$, while Krylov results are up to $N=20$. Inset shows the average standard error in $\lambda$ from the Krylov subspace method vs $N$ which is larger for smaller system sizes.