Traversability dynamics of minimal Sachdev-Ye-Kitaev Wormhole-inspired teleportation protocol with a parity-time ($\mathcal{PT}$)-symmetric non-Hermitian deformation

Technical Summary: Traversability Dynamics of Minimal SYK Wormhole-inspired Teleportation with PT-Symmetric Deformation

Problem Statement
Holography-inspired teleportation protocols, modeled by coupled Sachdev-Ye-Kitaev (SYK) systems, have established a framework for simulating traversable wormholes in quantum many-body systems. However, existing protocols typically assume closed-system dynamics governed by unitary, Hermitian Hamiltonians. Realistic quantum systems are open, interacting with environments that introduce gain and loss, leading to non-Hermitian dynamics. The central problem addressed in this work is to investigate how Parity-Time (PT) symmetric non-Hermitian deformations—specifically balanced gain and loss terms—affect the traversability, stability, and fidelity of wormhole-inspired teleportation. The authors seek to determine if such deformations can enhance signal transmission, induce phase transitions, or alter the causal structure of the emergent spacetime geometry.

Methodology
The study employs a theoretical framework based on the SYK model, which serves as a holographic dual to Jackiw-Teitelboim gravity. The system consists of two coupled SYK clusters (Left and Right), each containing $N=6$ Majorana fermions, prepared in a Thermofield Double (TFD) state.

• Hamiltonian Construction: The effective Hamiltonian is defined as $H_{eff} = H_L + H_R + g\hat{V} + H_{PT}$, where $g\hat{V}$ represents the standard Hermitian double-trace coupling enabling traversability, and $H_{PT} = i\gamma(Z_L - Z_R)$ introduces the PT-symmetric non-Hermitian deformation. Here, $\gamma$ represents the strength of balanced gain (on the Left) and loss (on the Right).
• Spectral Analysis: The authors utilize exact diagonalization to analyze the complex energy spectrum of $H_{eff}$. They focus on the behavior of eigenvalues as $\gamma$ increases, specifically looking for the coalescence of eigenvalues and eigenvectors at Exceptional Points (EPs).
• Statistical Characterization: To account for the inherent disorder of the SYK model (random all-to-all couplings), the authors perform a statistical study over 100 independent disorder realizations. They analyze the distribution of the critical non-Hermiticity threshold ($\gamma_c$) where PT symmetry breaks.
• Dynamical Simulation: The teleportation fidelity ($F$) and the success probability (norm of the teleported signal) are computed as functions of time ($t$), coupling strength ($g$), and non-Hermiticity ($\gamma$). The study examines the interplay between the scrambling time and the amplification effects.

Key Contributions and Results

1. Spectral Phase Transition and Exceptional Points:
   The system exhibits a sharp phase transition driven by spectral topology. In the exact PT-symmetric phase ($\gamma < \gamma_c$), the energy spectrum remains purely real, characterized by "level attraction" where resonant energy levels approach each other. At a critical threshold $\gamma_c$, these levels coalesce at a second-order Exceptional Point (EP). Beyond this threshold ($\gamma > \gamma_c$), the system enters the PT-broken phase, where eigenvalues bifurcate into complex conjugate pairs ($\lambda = \bar{E} \pm i\Gamma$), corresponding to distinct gain and loss modes.

2. Statistical Distribution of Stability:
   Due to the chaotic nature of the SYK spectrum, the critical threshold $\gamma_c$ is not a fixed constant but a stochastic variable. The authors find that $\gamma_c$ follows a log-normal distribution across disorder realizations. This indicates that while level repulsion generally protects the system, the specific breakdown point is hypersensitive to the microscopic level spacing and coupling matrix elements of individual realizations.

3. Causal Amplification:
   The PT-broken phase acts as a "Causal Amplifier." The non-Hermitian term induces an exponential growth in the norm of the teleported signal ($P_{success} \propto e^{2\gamma t \delta_{max}}$). Crucially, this amplification respects the causal structure of the emergent spacetime. The scrambling time $t^*$ (the time window for traversability) remains invariant under the deformation. The gain mechanism amplifies the signal magnitude without shifting the temporal location of the transmission window, effectively acting as an active medium within a fixed geometry rather than deforming the metric itself.

4. Gain-Induced Purification:
   Deep in the broken phase ($\gamma \gg \gamma_c$), the protocol exhibits a "Purification" effect. The non-Hermitian evolution acts as a spectral filter, suppressing thermal noise components (loss modes) and projecting the system onto the optimal teleportation subspace (the gain mode). This results in the teleportation fidelity plateauing near unity ($F \approx 1$) for post-selected states, rendering the protocol robust against detuning in the geometric coupling strength $g$.

Significance and Claims
The paper claims that non-Hermitian topology can be harnessed to enhance holographic quantum communication. By introducing balanced gain and loss, the authors demonstrate a mechanism to amplify signals in minimal quantum many-body systems without violating the causal connectivity of the emergent spacetime. The work suggests that the PT-broken phase provides a robust mechanism for signal recovery and entanglement distillation in noisy environments. The findings link the stability of holographic traversability directly to non-Hermitian spectral topology, showing that while the wormhole is topologically protected in the exact phase, it becomes dynamically unstable (amplifying) in the broken phase, offering a new control parameter for quantum information transport. The study is presented as a proof-of-principle demonstration using finite $N=6$ systems, arguing that the observed mechanisms (EP topology, causal amplification, and purification) are fundamental principles expected to persist in larger or experimental implementations.