Quantum Error Correction in SYK and Bulk Emergence

TL;DR

The paper investigates how quantum error correction properties in the SYK model encode bulk emergence and holographic features. By computing the price of reconstructing time-evolved fermionic operators via modular-flow correlators and replica techniques, it shows that SYK at large N exhibits maximal modular chaos and that the resulting quantum extremal surface location matches predictions from a simple JT gravity bulk model. The results establish a concrete link between operator size, modular Hamiltonians, and entanglement‑wedge reconstruction, and demonstrate an emergent Type III1 von Neumann algebra via half‑sided modular inclusion. Together, these findings illuminate how bulk locality and gravitational diagnostics can arise from non-spatial degrees of freedom in a chaotic quantum many-body system.

Abstract

We analyze the error correcting properties of the Sachdev-Ye-Kitaev model, with errors that correspond to erasures of subsets of fermions. We study the limit where the number of fermions erased is large but small compared to the total number of fermions. We compute the price of the quantum error correcting code, defined as the number of physical qubits needed to reconstruct whether a given operator has been acted upon the thermal state or not. By thinking about reconstruction via quantum teleportation, we argue for a bound that relates the price to the ordinary operator size in systems that display so-called detailed size winding of Nezami et al. (2021). We then find that in SYK the price roughly saturates this bound. Computing the price requires computing modular flowed correlators with respect to the density matrix associated to a subset of fermions. We offer an interpretation of these correlators as probing a quantum extremal surface in the AdS dual of SYK. In the large $N$ limit, the operator algebras associated to subsets of fermions in SYK satisfy half-sided modular inclusion, which is indicative of an emergent Type III$_1$ von Neumann algebra. We discuss the relationship between the emergent algebra of half-sided modular inclusions and bulk symmetry generators.

Figures (4)

• Figure 1: We imagine that we have a system of qubits prepared in their thermo-field double state, whose entanglement is represented by the dashed lines. We will ask the question of what is the largest number of qubits, $K$, from the right that we can give to the left such that we can just reconstruct an excitation on $r = \overline{LK}$.
• Figure 2: We illustrate the putative bulk picture that we test in Sec. \ref{['sec:bulkpicture']}. We imagine a bulk with $N$ free fermions of mass set by the boundary dimensions of the SYK fermions in the IR limit. We then imagine finding a QES in the bulk associated to $L \cup K$ by fixing a small interval $\boldsymbol{i}$, associated with the $K$ fermions on the right, and then extremizing the generalized entropy for $\ell \cup i$ where only the $K$ fields are included in the entropy for $\ell \cup i$. The remaining $N-K$ fields only contribute to the entropy in $\ell$.
• Figure 3: The top figure illustrates the path integral one must do to prepare the pure state density matrix $\ket{\beta}\bra{\beta}$. The bottom two figures illustrate tracing out subsystems from the TFD to produce a mixed state density matrix. In Fig. \ref{['fig:rhoLK']}, we have traced out $r$ to produce $\rho_{LK}$, and in Fig. \ref{['fig:rhor']} we have traced out $LK$ to produce $\rho_r$. The dashed lines remind us that the two "wires" (contours) are coupled via Euclidean evolution with the SYK Hamiltonian. Thus, they tell us what parts of each wire have the same Euclidean time. Contours of Euclidean time length $\beta/2$ have been indicated. $L_r$ and $L_K$ refer to the image of $r$ and $K$ under Euclidean evolution by $\beta /2$.
• Figure 4: On the left we illustrate the correlator $\bra{\beta} \psi_L(\tau) (\rho_{LK}^2 \otimes \rho_r) \psi_R(\tau')\ket{\beta}$. Since only the topologies of the wires and the dashed lines are meaningful, we can write this diagram as on the right, where it looks like a standard replica contour. Black dots connected by dashed lines are at the same Euclidean time.