WITP: Wormhole-Inspired Teleportation Protocol

• WITP is a quantum protocol that uses a Thermofield Double state and k-local scrambling to mimic wormhole traversal for teleporting quantum states.
• It employs classical measurement and a conditional unitary correction to recover the teleported state, reflecting the ER=EPR duality in holography.
• The protocol’s reduced circuit complexity via the switchback effect enables experimental simulation of emergent wormhole dynamics in quantum systems.

Wormhole-Inspired Teleportation Protocol (WITP) is a quantum information protocol motivated by and directly tied to the holographic correspondence between maximally entangled states and traversable Einstein–Rosen bridges (ER=EPR). In its most studied form, WITP operates by first preparing two large quantum systems in a Thermofield Double (TFD) state, representing maximal entanglement between two “boundary” systems. A quantum state to be teleported (“teleportee”) is injected into one side and rapidly scrambled via a k-local, chaotic unitary evolution. Subsequent measurement on a designated subsystem, followed by transmission of the classical outcome to the remote side, enables recovery of the quantum information—mirroring the process of a probe traversing the interior of a wormhole, made traversable via double-trace or equivalent interboundary coupling. The protocol provides a profound operational realization of ER=EPR and underscores the geometrization of quantum information transfer.

1. Structural Elements of the Protocol

WITP is built on the following sequence of operations:

1. Entangled Mediators: Two subsystems, commonly designated as “Alice” (A) and “Bob” (B), are initialized in a maximally entangled state. Operationally, for N qubits, this is the TFD state:

$$|\mathrm{TFD}\rangle = \sum_{I} |I_A\rangle \otimes |I_B\rangle = (|00\rangle + |11\rangle)^{\otimes N}$$

Here, $|I_A\rangle$ ($|I_B\rangle$) form complete bases for Alice (Bob).

1. Insertion and Scrambling: A teleportee qubit in state $$|\Phi_T\rangle = \Phi(0)|0_T\rangle + \Phi(1)|1_T\rangle$$ is merged with Alice's share. The system $A$ plus teleportee is then evolved by a scrambling unitary $V$, typically $k$-local with a scrambling time $t_* \approx \log N$. After scrambling,

$$V|{\rm initial}\rangle = \sum_{I, k, \theta, \alpha} \Phi(k) V^{(\theta, \alpha)}_{kI} |I_B\rangle \otimes |\theta, \alpha\rangle$$

The basis on $A$ plus teleportee is split into a “small” subsystem $\theta$ (e.g., $2$ qubits for $1$-qubit teleportation) and a “large” remainder $\alpha$.

1. Measurement and Classical Transmission: Alice measures her $\theta$ subsystem in the computational basis, obtaining outcome $\theta$ as a random bitstring. This outcome is sent classically to Bob.
2. Conditional Unitary Correction: Bob applies a state-dependent unitary $Z^\theta$ on his mediator subsystem, defined through

$$Z^\theta_{I, \beta j} = \sum_{\gamma} V^{\dagger (\theta, \gamma)}_{I, j} W_{\gamma \beta}$$

with $W_{\alpha \beta} = V^\dagger_{\alpha\beta}$ yielding a circuit of complexity $O(N)$ via the switchback effect; in contrast, a naive correction would be $O(N\log N)$ in depth.

1. Teleported State Recovery: After Bob's correction, his qubit subsystem emerges in the state

$$|\Phi_B\rangle = \Phi(0)|0_B\rangle + \Phi(1)|1_B\rangle$$

while the remaining mediator qubits $(\alpha,\beta)$ remain maximally entangled.

The protocol formalizes quantum teleportation as a process in which quantum information traverses the “interior” of a wormhole furnished by entanglement, with scrambling representing the in-fall of information and the nonlocal correction standing for the traversability-inducing interaction.

2. Holographic Interpretation via ER=EPR

The WITP has a precise dual realization in holography:

• The TFD state is dual to an eternal black hole featuring a non-traversable Einstein–Rosen bridge.
• Scrambling of the teleportee into Alice's subsystem corresponds to the infalling matter entering the left-side black hole.
• The classical transmission of $\theta$ and the corresponding application of $Z^\theta$ represent a double-trace deformation of the Hamiltonian, coupling the two boundaries at a specific time.

The effective unitary encoding the full protocol is:

$$\mathcal{U} = \sum_\theta \Pi_\theta(t_*) \otimes S^\theta(t)$$

with $S^\theta(t) = U^\dagger(t) Z^\theta U(t)$ and $U(t)$ time-evolution. In the gravitational picture, the double-trace deformation generates a negative energy shockwave,

$$\mathcal{U}_\mathrm{GJW} = \exp\{ i g \mathcal{O}_L\mathcal{O}_R \}$$

inducing a horizon shift

$\alpha \propto E e^{|t|}$

of negative sign, thereby permitting traversability for the infalling state. The protocol thus makes direct use of the Shenker–Stanford analysis of shockwave-induced null shifts in black hole spacetimes.

Classical data $\theta$ carries no information about $\Phi(0), \Phi(1)$ and simply selects the recovery unitary; all quantum information is transported through the entangled bulk, never leaking to the environment.

3. Role of Scrambling, Complexity, and Switchback Effect

The scrambling operator $V$ is critical, dictating that the teleportee’s information is thoroughly mixed among Alice’s $N$-qubit system:

• In k-local models, scrambling is achieved in $t_* \sim \log N$ (scrambling time).
• The measurement on subsystem $\theta$ selects a random matrix element of $V$; Bob’s recovery requires effectively “inverting” the right pattern of scrambling.

By leveraging the switchback effect, the required circuit to implement $Z^\theta$ (undoing the scrambling up to local rotations) is only $O(N)$ in depth — a substantial reduction from the naive $N \log N$ complexity, making the protocol feasible for simulation in large-$N$ quantum systems or complex quantum circuits.

4. Teleportee Memory and Observability

A salient feature of WITP is that the teleported system may bear “memory” of its passage through the wormhole:

• If the initial TFD mediator is perturbed (for example, by injecting an extra photon on Alice’s side before scrambling), the shockwave generated will interact with the teleportee inside the wormhole.
• Upon emergence, the teleportee state will carry signatures—quantum correlations—of this encounter, potentially accessible via measurement.
• In the quantum circuit language, memory is encoded as nontrivial correlations established by the composition of $V$ and $V^\dagger$, or more generally, by the scrambling evolution and its partial inversion.

This is consistent with the dual bulk physics, wherein a traveller through the wormhole can be affected by bulk interactions; such effects should be, in principle, observable within sufficiently controllable quantum simulation experiments.

5. Operational and Experimental Realization

The protocol is operationally implementable with quantum computing platforms that can:

• Realize large, entangled qubit registers (the TFD or an equivalent entangled resource state).
• Engineer k-local or highly nonlocal unitary evolutions to model scrambling.
• Select subsystems for measurement and reliably perform post-selection.
• Transmit measurement data (the classical string $\theta$) to the remote side.
• Execute conditional logic based on classical input to implement $Z^\theta$ operations.

Experimental proposals include simulating “shells” (as stand-ins for boundary quantum systems) using separate quantum computers or engineered qubit arrays. The protocol's complexity is limited by the scrambling and recovery steps, but thanks to the circuit simplifications from the switchback effect, realization with system sizes ($N \sim 10^2$) is projected to be within reach. Notably, such setups would not only demonstrate high-fidelity quantum teleportation but, if the initial state is appropriately perturbed, could provide direct experimental signatures of the induced wormhole geometry (for example, via bulk shockwave effects).

6. Generalizations, Theoretical Impact, and Fundamental Consequences

WITP extends quantum teleportation beyond the regime of simple two-qubit EPR pairs, embedding the process into the nonperturbative bulk-boundary dictionary of holography:

• The transport channel is a traversable wormhole, realized concretely by mediator entanglement and boundary-implemented scrambling/coupling dynamics.
• Classical communication remains a necessary element but contains no quantum information; all sensitive data transits the “wormhole” channel encoded in entanglement and correlators.
• The protocol underlines the operational duality between quantum circuits simulating strongly coupled dynamics and geometric bulk connectivity in gravity.
• Its bulk realization via negative-energy shocks links quantum information protocols directly to violation of average null energy conditions, core to traversable wormhole physics.

This synthesis bridges quantum information theory and gravitational concepts, lending concrete protocol-level content to the ER=EPR conjecture, and bears direct implications for the simulation or even observation of emergent spacetime phenomena using quantum many-body systems.

7. Summary Table of Key Components

Element	Quantum Circuit/Protocol	Gravitational Dual
Mediator (TFD state)	Maximally entangled N qubits	Non-traversable Einstein–Rosen bridge
Scrambling	k-local unitary V	Infall into black hole/ERB
Classical message θ	Measurement outcome	Boundary-coupling (double-trace)
Recovery unitary Z^θ	Corrects scrambled info	Negative energy shockwave, opens wormhole
Memory effect	Interference via V, V†	Shockwave imprint, interactions in bulk

The protocol not only provides a pathway for teleporting quantum information with protection via bulk geometry analogues but also illuminates how classical communication and quantum error correction interleave with nonperturbative gravitational physics. Experimental implementation in quantum simulators promises to further clarify these deep connections.