SYK Model-Based Teleportation Protocol

Updated 21 August 2025

SYK model-based teleportation protocols are quantum schemes that harness chaotic Majorana fermions to transmit quantum states via traversable wormhole dynamics.
The method employs a sequence of SWAP operations, bilinear couplings, and extraction steps to encode, couple, and retrieve quantum information with near-perfect fidelity.
Crucial diagnostics, including quantum mutual information and correlator growth, link holographic duality with operator scrambling and topological order insights.

A SYK model-based teleportation protocol is a quantum information transmission scheme leveraging the dynamical properties of the Sachdev–Ye–Kitaev (SYK) model—an ensemble of Majorana fermions with all-to-all random interactions. These protocols exploit connections between quantum chaotic dynamics, holographic duality, traversable wormhole physics, and quantum teleportation, providing an explicit mechanism for robust, high-fidelity transfer of quantum states across highly entangled many-body systems. The implementation and analysis of these protocols has advanced the understanding of information retrievability, emergent gravitational features, operator growth, and topological order in the context of strongly interacting quantum matter.

1. Foundations: SYK Model, Thermofield Double States, and Wormhole Duality

The SYK model consists of $N$ Majorana fermions interacting via $q$ -body random all-to-all couplings. At large $N$ and low temperatures, it is maximally chaotic and believed to be dual to nearly-AdS $_2$ gravity. The protocol utilizes two copies of the SYK model (“left” $l$ and “right” $r$ ) prepared in a highly entangled thermofield double (TFD) state:

$|\mathrm{TFD}\rangle = \frac{1}{\sqrt{Z}} \sum_n e^{-\beta E_n/2} |E_n\rangle_l \otimes |E_n\rangle_r$

The TFD state exhibits maximal entanglement and, in holographic duality, corresponds to a pair of black holes joined by a nontraversable Einstein–Rosen bridge.

The crucial modification is the introduction of a weak, instantaneous bilinear coupling $U = e^{i\mu V}$ (typically, $V = i \sum_j \psi^j_l \psi^j_r$ at $t=0$ ). In the gravitational dual, this renders the wormhole traversable, allowing information injected on one side to emerge on the other as a bulk excitation traversing the bridge (Gao et al., 2019).

2. Protocol Architecture: Injection, Coupling, and Extraction

The teleportation sequence is typically executed in three steps:

Insertion: At time $t=-t'$ , a message qubit $Q$ is swapped into a Dirac fermion degree of freedom of the left SYK system via a “simple” SWAP gate acting only on the chosen fermion pair and the qubit. In the Majorana basis, Dirac fermions are constructed from pairs of Majorana operators, and the SWAP is represented using matrix elements in Pauli or fermionic algebra.
Wormhole Coupling: At $t=0$ , the unitary $U=e^{i\mu V}$ is enacted, coupling the left and right systems without requiring complex nonlocal operators. The construction is such that (upon expansion) it can be viewed as a measurement on the left followed by a corresponding unitary on the right, and is experimentally equivalent to a classical-communication channel variant.
Extraction: At $t=+t'$ , a matching SWAP operation on the right system extracts the message, mapping the encoded state onto target qubit $T$ . The correlation between a reference qubit $R$ (originally entangled with $Q$ before insertion) and $T$ is then evaluated to diagnose successful teleportation.

In the improved version, the SWAP is generalized to composite operators built from multiple Majoranas, enhancing the fidelity by encoding the logical qubit into a more complex Hilbert subspace (Gao et al., 2019).

3. Mathematical Framework and Fidelity Diagnostics

The fidelity of teleportation is quantified via the quantum mutual information $I_{RT}$ between $R$ and $T$ , and the reduced density matrix $\rho_{RT}$ . For perfect teleportation,

$I_{RT} = S(R) + S(T) - S(RT) = 2 \log 2$

The essential correlators reduce to expectation values such as $\langle \chi_l\chi_l^\dagger U^\dagger \chi_r\chi_r^\dagger U \chi_l\chi_l^\dagger \rangle$ , where $\chi_{l,r}$ are Dirac annihilation/creation operators.

A critical diagnostic is the two-point correlator

$K(t, t') = \langle \{\psi_l(-t'), U^\dagger \psi_r(t) U\} \rangle$

In the low-temperature, large- $q$ semiclassical regime, the correlator exhibits exponential growth governed by the Lyapunov exponent $2\pi/\beta$ , and approaches an ideal peak at the causal time for transmission through the wormhole. The density matrix elements for $RT$ are explicitly calculated and arranged in block form, with perfect fidelity achieved when $|_{14}| = 1$ and $_{11} = 1$ .

Enhanced protocols utilize redundancy—encoding the message in a product of $q/2+1$ Majorana fermions or $p \gg q$ in the high-temperature regime. In the large- $q$ limit, the protocol's mutual information approaches its maximal value.

4. Physical Interpretation: Traversable Wormholes and Size Winding

The gravitational dual of the SYK teleportation protocol is the transmission of quantum information through a semiclassical traversable wormhole. The instantaneous left-right interaction $U$ triggers a negative average null energy in the gravity picture, opening the wormhole. The teleportation protocol reproduces key holographic signatures:

A sharp “time window” for signal transfer
Fidelity and causality determined by the wormhole geometry
Exponential growth in correlators corresponding to scrambling

On the operator level, the protocol is governed by “size winding,” a linear phase accumulation in the expansion of a scrambled Majorana operator in terms of operator size. This phenomenon is critical—the left-right coupling $e^{i\mu V}$ imparts a size-dependent phase that “unwinds” the operator, facilitating information recovery on the right side (Lykken et al., 13 May 2024). The mathematical representation aligns with the structure of boost parameters in nearly-AdS $_2$ gravitational dynamics.

5. Operational Simplicity, Error Protection, and Generalizations

A notable feature is the use of “simple” operators: all components—SWAPs, couplings, and measurements—are constructed out of Pauli matrices, Dirac or Majorana bilinears, and local projectors. This distinguishes SYK-based protocols from those requiring intensive decoding of maximally scrambled operators (“precursors”).

The protocol encodes the logical qubit as the occupation number of a Dirac fermion (or generalized composite operator), providing high resistance against local errors and boundary perturbations (error protection). When generalizing to composite-fermion encodings, the scheme achieves near-perfect fidelity, especially at low temperatures where gravitational features dominate.

The scheme admits a classical channel version, wherein measurement outcomes are communicated classically and feedforward corrections are applied, with essential holographic features preserved. This variant increases feasibility in quantum network implementations (Lykken et al., 13 May 2024).

6. Broader Context: Comparison, Topological Order, and Future Directions

SYK model-based teleportation protocols are benchmarked against analogous schemes in non-holographic systems:

In the “commuting SYK” model, integrable dynamics with peaked size distribution provide pseudo-holographic features, yet true holographic scrambling and exponential growth are absent (Gao, 2023).
Local models, such as the 2D spin-$1/2$ XY chain with “rainbow scar” eigenstates, construct alternative many-body teleportation protocols that replace SYK's chaotic scrambling with structured long-range entanglement and iterative measurement/feedback (Agarwal et al., 2022).

Beyond dynamical features, recent research reveals that long-range teleportation protocols inherently require resource states with symmetry-protected topological (SPT) order. For “standard” teleportation schemes employing Pauli-conditioned corrections, the resource state belongs to a nontrivial SPT phase with Abelian protecting symmetry $\mathcal{G}_k = (\mathbb{Z}_2 \times \mathbb{Z}_2)^k$ , characterizable by nonlocal string order parameters (Hong et al., 2023). This topological framework unifies the understanding of both stabilizer-based and nonstabilizer resource states required for protocol robustness.

Future directions include:

Extensions to measurement-induced protocols and cSYK models (complex fermions), where projective measurements combined with decoding gates can trigger phase transitions in the quantum extremal surface and alter the entanglement wedge, affecting teleportation phases (Brinster et al., 10 Jan 2025).
Quantum circuit simulations investigating the interplay of temperature ( $\beta$ ), swap location, operator distance, random matrix structure, and multi-qubit fidelity measures (e.g., stabilizer-based fidelity for Bell pairs) (Joshi et al., 18 Jun 2025).
Exploration of partial scrambling, non-holographic analogs, and experimental testbeds for probing traversable wormhole phenomena in laboratory-controlled settings.

7. Summary

SYK model-based teleportation protocols realize quantum information transfer in maximally chaotic, strongly entangled systems via a sequence of local, physically implementable operations. The protocols concretely instantiate the deep links between many-body quantum chaos, holographic wormhole dynamics, operator scrambling (especially the concept of size winding), and topological order manifested as SPT phases in the resource state. The methodology is adaptable (including classical channel regimes and multi-qubit encodings) and provides an operative route to simulate and probe fundamental aspects of quantum gravity, information retrieval, and condensed matter dynamics in a unified framework.