Quantum magic is necessary but not sufficient for wormhole-inspired teleportation

Abstract: We investigate the dynamics of Quantum magic, formally known as non-stabilizerness, quantified by the stabilizer Rényi entropy (SRE), across the stages of the wormhole-inspired teleportation protocol (WITP) in the Sachdev-Ye-Kitaev (SYK) model. By tracking the SRE of the full pure state across scrambling, message insertion, left-right coupling, and right-side extraction, we uncover a regime-dependent relationship between magic accumulation and teleportation fidelity. In the gravitational (low temperature) regime, fidelity rises concurrently with magic from early times, whereas in the peaked-size (high temperature) regime, the magic saturates near the Haar-typical value before teleportation onset. A baseline-subtracted diagnostic comparing coupled and uncoupled protocols reveals that the double-trace coupling first suppresses and then channels non-stabilizer resources toward the teleportation signal, with the channel amplitude decreasing monotonically with inverse temperature. Comparison with a chaotic random two-local model, which generates near-maximal magic yet fails to teleport, demonstrates that structured magic redistribution, rather than raw non-stabilizerness, underlies successful wormhole traversal. Moreover, the magic transiently dips at the fidelity peak, marking the teleportation event in the time domain. Our results are robust across the three system sizes studied ($N_{\mathrm{maj}}=8,10,12$), and the fidelity-magic trajectories exhibit an approximate collapse across system sizes when the SRE is normalized by the Haar-typical prediction.

• The paper demonstrates that quantum magic is necessary but not sufficient for high-fidelity wormhole-inspired teleportation, emphasizing the role of structured resource redistribution.
• The paper employs the stabilizer Rényi entropy within the SYK model and controlled scrambling protocols to quantify non-stabilizerness at various teleportation stages.
• The paper shows that only models with coherent, structured magic, as seen in the SYK regime, achieve non-classical teleportation unlike chaotic or integrable comparators.

Quantum Magic as an Operational Resource in Wormhole-Inspired Teleportation

Introduction

This work addresses the operational role of quantum "magic"—the non-stabilizerness of quantum states characterized via the stabilizer Rényi entropy (SRE)—during wormhole-inspired teleportation protocols (WITP) in the Sachdev-Ye-Kitaev (SYK) model. The SYK model provides an analytically tractable platform to study quantum chaos, holography, and the emergence of gravitational dynamics in strongly coupled quantum matter. Specifically, at low temperatures, SYK displays maximally chaotic dynamics and is believed to have a gravitational dual, including traversable wormhole solutions. Implementing WITP in such models, where quantum information is effectively teleported between two entangled systems via a double-trace coupling, provides a testbed for studying the dynamical use of non-stabilizer resources in teleportation.

The central question posed is: Is quantum magic, a resource that distinguishes universal quantum computing from classically simulable stabilizer circuits, necessary or sufficient for successful wormhole-inspired teleportation? This work rigorously explores this question by tracking the SRE across all stages of the protocol.

Protocol and Diagnostic Framework

The paper considers the standard SYK-based WITP, involving sequences of controlled scrambling, information encoding, double-trace coupling (auxiliary to the traversable wormhole connection), and right-side message extraction. The Hilbert space is constructed from pairs of Majorana fermions mapped onto qubits, starting from a thermofield double (TFD) state at variable temperature.

Magic is quantified via the SRE:

$$M_2(\psi) = -\log_2 \left( \frac{1}{d} \sum_{P\in\mathcal{P}_n} |\langle\psi|P|\psi\rangle|^4 \right)$$

where $\mathcal{P}_n$ is the $n$-qubit Pauli group. The computation uses an optimized Walsh-Hadamard transform to enable evaluation for up to 12 qubits.

The protocol tracks magic explicitly across (i) TFD preparation, (ii) backward scrambling, (iii) message insertion, (iv) forward scrambling, (v) left-right coupling, and (vi) right-side extraction.

Critically, the analysis introduces a baseline-subtracted diagnostic:

$\delta M_2(t) = M_2^{g=g^*}(t) - M_2^{g=0}(t)$

measuring the coupling-induced (nontrivial) redistribution of non-stabilizer resources, controlling for the generic growth due to scrambling.

Comparison is made to (i) a non-holographic, integrable transverse-field Ising model (TFIM), and (ii) a chaotic random two-local ensemble (R2L) which provides near-maximal magic but is not holographically dual to gravity.

Main Results

1. Fidelity-Magic Dynamics Across Teleportation Regimes

A regime-dependent interplay between SRE and teleportation fidelity is established. In the gravitational (low $T$) regime, fidelity and SRE both rise, tracing out a looped parametric trajectory in $(F, M_2)$ space. This regime exhibits "size winding": operator growth is phase-correlated and highly structured, yielding a direct channel from non-stabilizerness to teleportation signal.

In contrast, at high temperatures (peaked-size regime), SRE saturates quickly near its Haar-typical value but fidelity remains at the classical limit ($F=1/4$) until a sharp onset once SRE is nearly maximized. Here, operator growth yields generically high magic, but teleportation only emerges once the system approaches near-Haar randomness, indicating a lack of structured channel for information transfer.

2. Structured Versus Unstructured Non-Stabilizerness

A chaotic R2L spin model generates near-maximal SRE under all-to-all two-local dynamics but fails to teleport (achieving only classical fidelity), demonstrating that raw non-stabilizerness is not sufficient for teleportation. Conversely, only SYK generates the necessary structured redistribution of magic to support non-classical fidelity. Integrable dynamics (TFIM) are incapable of generating sufficient magic or fidelity.

Analysis of the Pauli-weighted distribution of the SRE (Fig. 7 in the original) shows that, in SYK at the fidelity peak, non-stabilizer content is concentrated in a sharply defined set of operators that overlap with the teleportation channel, while R2L distributes magic uniformly with no such structure.

3. Temporal and Finite-Size Structural Features

In the peaked-size regime, SRE briefly dips at the fidelity maximum, indicating that successful teleportation transiently concentrates the available non-stabilizer resource onto the specific fidelity-relevant correlators. This feature is absent in the gravitational regime, where magic and fidelity increase together monotonically. Such protocol-level signatures are robust across system sizes ($N_\mathrm{maj}=8,10,12$).

Baseline-subtracted diagnostics confirm a two-phase structure: the coupling first suppresses magic (nearly-Clifford rearrangement) before channeling it toward the teleportation signal. The amplitude of this "magic boost" decreases monotonically with inverse temperature.

4. Universality and Haar Normalization

Normalizing $M_2$ by its Haar-typical value yields an approximate collapse of the fidelity-magic curve across system sizes, suggesting that fractional Haar saturation captures the leading system-size dependence of the teleportation resource structure. The contrast between gravitational and peaked-size regimes sharpens as $N_\mathrm{maj}$ increases, indicating stronger asymptotic distinctness.

Interpretations and Theoretical Implications

These findings support several theoretical implications:

• Necessity but Not Sufficiency: Non-stabilizerness is required for quantum teleportation in the SYK WITP, but generic magic accumulation is not sufficient; the spatial and operator structure of magic, shaped by the underlying chaotic dynamics, is essential.
• Protocol-Level Magic as Diagnostic: Magic dynamics encode information about the operational mechanism of teleportation beyond static or equilibrium entanglement measures. The SRE-fidelity trajectory distinguishes teleportation regimes even when traditional chaos/holography diagnostics (OTOCs, spectral statistics) are inaccessible due to finite-size limitations.
• Connections to Complexity: The structured redistribution mechanism resonates with the complexity-geometry conjecture in AdS/CFT, where only organized resource growth (not just raw metric or non-Cliffordness) is dual to meaningful bulk geometry or information flow.
• Experimental Accessibility: The protocol-level SRE and its channeling diagnostic are experimentally testable using randomized measurements, requiring only comparison runs to extract the operationally relevant effect.

Future Directions

Open problems include: (i) direct computation of operator size-winding and its coupling to magic dynamics at finite $\mathcal{P}_n$0, (ii) rigorous analysis of the universal collapse under Haar-normalization at larger $\mathcal{P}_n$1, and (iii) extension of the magic diagnostics to more complex holographic circuits and beyond exact diagonalization via efficient sampling or tensor-network methods.

Conclusion

This work conclusively demonstrates that while quantum magic is indispensable for wormhole-inspired teleportation, it is the coherent, dynamically structured redistribution—rather than the absolute amount—of non-stabilizerness that constitutes the true operational resource. These insights clarify the distinct nature of holographic and non-holographic quantum channels and establish protocol-resolved SRE analysis as a powerful tool for diagnosing gravitationally-inspired quantum processes.