Remote Gottesman-Kitaev-Preskill Entanglement Generation between Trapped Ions

The gist The protocol proposes a heralded method for generating remote entanglement between Gottesman–Kitaev–Preskill qubits encoded in trapped ion motional modes by using internal electronic states to interface these motional qubits with emitted photons.

How it works

1. The protocol involves a three-step entanglement distribution process where the local operations are detailed in Figure 2 In Step 1, each node prepares a motional GKP logical state ¯0⟩ and entangles it with the internal state of the ion as (¯0⟩ g0⟩ + ¯1⟩ g1⟩)/√2 In Step 2, excitation and subsequent photon emission from the internal state convert this local motion–ion entanglement into a motion–ion–photon Greenberger-Horne-Zeilinger (GHZ) state The motional GKP qubit does not directly couple to a photon, so the internal state mediates the interface between the motional qubit and the photonic qubit In Step 3, the emitted photons from the two nodes are sent to a BSM node Conditioned on a successful Bell-state measurement, the internal states are measured in a Pauli-X basis, which disentangles the internal states and the two motional modes resulting in the GKP Bell state.

Noise Models

The paper analyzes several imperfections related to motional GKP states, including photon recoil, motional heating, and motional dephasing The noise channels are summarized in Figure 3 Photon loss is modeled as amplitude damping within the vacuum–one-photon qubit space for the single-rail encoding, with the channel modeled by Lηph(ρ) = Eˆ0ρEˆ†0 + Eˆ1ρEˆ†1 Photon recoil during Step 2 is represented by the operator Rˆ(t) = D̂[-iηeiωmt]D̂(iη) Motional heating is modeled as a Gaussian random-displacement channel Eheat(ρ) = 1/πnt¯˙ fly Zd2α e−α2/nt¯˙ fly D̂(α)ρD̂†(α) Motional dephasing is modeled by Gaussian phase diffusion Edeph(ρ) = 1/p2πκϕtfly Z∞−∞dϕ exp −ϕ2/2κϕtfly e iφnˆρe−iφnˆ.

Evaluation and Results

The evaluation uses stabilizer subsystem decomposition (SSSD) to compute the logical state ρL obtained by ideal GKP decoding The logical Bell-state fidelity FL is computed following the manner of Refs. [68, 69] Under noiseless conditions, the decoded logical fidelity decreases monotonically with increasing GKP squeezing level ∆dB At 6 dB, the decoded logical fidelity is FL = 91.78%, whereas at 15 dB it reaches FL = 98.82%.

Effect of Physical Noise

Photon loss acts as amplitude damping within the vacuum–one-photon qubit space for the single-rail encoding, and its contribution to infidelity increases with ploss For the polarization encoding, photonic qubits lie in the one-photon subspace, and photon-loss events are discarded at BSM. Photon recoil during spontaneous emission does not preclude high-fidelity remote GKP Bell-state generation when the Lamb–Dicke parameter is small The recoil-induced degradation is suppressed when the Lamb–Dicke parameter is small, and the photon wavepacket duration is short compared to the motional period. Motional heating and dephasing both distort the GKP grid and reduce the distinguishability of logical GKP lattice points For representative flight time and trap parameters, the generated remote GKP Bell state remains above 95% logical fidelity when the motional heating rate is below the order of 102 quanta/s and the motional dephasing rate is below the order of 10 Hz.

Conclusion

The protocol successfully generates remote entanglement between GKP qubits encoded in trapped ion motional modes by interfacing them with emitted optical photons using internal electronic states The two photonic encodings exhibit similar recoil sensitivity in the weak-recoil regime but differ in the strong-recoil regime. Photon loss affects the single-rail encoding by introducing false heralds, whereas for the polarization encoding, photon-loss events are rejected by the two-photon detection condition. The calculations provide a link-level characterization of the proposed remote motional-GKP entanglement protocol over experimental noise parameters. Several extensions remain for future work, such as modeling other bosonic codes and including finite control errors in the complete entanglement-generation sequence.

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Remote Gottesman-Kitaev-Preskill Entanglement Generation between Trapped Ions

Haruki Aoki,1,∗ Kazufumi Tanji,2 Shohei Kiryu,2 Masahiro Takeoka,2, 3, † and Junko Ishi-Hayase1,2 3 (Dated: October 16, 2026)

Distributing entanglement directly between error-corrected logical qubits is an important capability for scalable quantum networks. We propose a heralded protocol for generating remote entanglement between Gottesman–Kitaev–Preskill qubits encoded in the motional modes of spatially separated trapped ions, using the internal electronic states to interface the motional qubits with emitted photons. We derive the protocol for single-rail and polarization photonic encodings and evaluate the effects of finite GKP squeezing, photon loss, photon recoil, motional heating, and motional dephasing on the decoded logical state. Moreover, we numerically analysed the fidelity of distributed logical Bell-pair fidelity. We find that one can achieve more than 95% of fidelity in the realistic situation. The protocol combines motional-control and photonic-networking operations that have already been demonstrated in trapped-ion experiments.

I. INTRODUCTION

Entanglement between remote physical systems is a fundamental resource for quantum networks with quantum repeater [1–3], enabling applications such as distributed quantum computation [4–6], and distributed quantum sensing [7–10].When two remote systems cannot interact directly, entanglement swapping is a standard way to generate entanglement: each stationary qubit is first entangled with a flying photonic qubit, and a Bell-state measurement (BSM) on the photons heralds entanglement between the remote stationary qubits. Proof-of-principle experiments have been reported in various stationary qubit platforms, including trapped-ion systems [11–20], cold atoms [21–28], and solid-state electron spin [29–36]. Despite these recent experimental progress, fidelity of the generated entanglement is still limited by qubit decoherence and by errors induced during photon generation. The dominant sources of stationary-qubit decoherence vary across physical platform; including magnetic field fluctuation in trapped ions [37], laser fluctuations of optical tweezers in neutral atoms [38], and phonon or spin baths in solid-state spin systems [39]. In addition to such stationary-qubit decoherence, generation of qubit-photon entanglement may induce errors, for example, photon recoil [18, 40–42]. Together, stationary-qubit decoherence and photon generation errors impose a practical limitation of fidelity on the generated remote entanglement. To overcome this limitation, the quantum error correction code (QECC) has been explored [43, 44]. Alternatively, QECC can impove fidelity deterministically∗ aoki.2.71828@keio.jp † takeoka@elec.keio.ac.jp ‡ hayase@appi.keio.ac.jp if physical operations and raw Bell state fidelity surpass certain threshold[45]. Therefore, high fidelity Bell pair can be efficiently obtained by using QECC [45]. QECCs can be broadly classified into discrete-variable (DV) codes and bosonic codes which encode information in continuous-variable (CV) degree of freedom. DVQECC schemes are deterministic and thus require less remote Bell pairs than distillation.

1. The AI system can generate remote entanglement between error-corrected logical qubits encoded in motional modes of spatially separated trapped ions by utilizing internal electronic states to interface motional qubits with emitted photons, as proposed by the protocol combining motional-control and photonic-networking operations that have already been demonstrated experimentally.

2. The system can achieve a logical Bell-state fidelity exceeding 95% in realistic situations by incorporating stabilizer subsystem decomposition [63, 64] to reconstruct the logical state from expectation values of logical Pauli operators PˆL,i ∈ n ˆI, X, ˆ Y, ˆ Zˆo.

3. The AI system can be optimized for specific encoding schemes: a single-rail encoding will show reduced fidelity under photon loss because photon loss acts as amplitude damping within the vacuum–one-photon qubit space, whereas the polarization encoding maintains fidelity because the BSM accepts the twophoton detection pattern and photonic qubits lie in the one-photon subspace.

4. The system can be robust against motional noise by utilizing noise models that characterize photon recoil, motional heating, and motional dephasing, allowing for quantification of degradation over experimental parameter ranges where fidelity remains above 95% under conditions like nt¯˙ fly = 300 quanta/s and κϕ = 30 Hz.

5. The AI system can adapt its entanglement generation strategy based on noise regimes, as the analysis shows that photon recoil during spontaneous emission does not by itself preclude high-fidelity remote GKP Bell-state generation when the Lamb-Dicke parameter is small, suggesting fast spontaneous photon emission is preferable.

Distributing entanglement directly between error-corrected logical qubits is an important capability for scalable quantum networks. We propose a heralded protocol for generating remote entanglement between Gottesman--Kitaev--Preskill qubits encoded in the motional modes of spatially separated trapped ions, using the internal electronic states to interface the motional qubits with emitted photons. We derive the protocol for single-rail and polarization photonic encodings and evaluate the effects of finite GKP squeezing, photon loss, photon recoil, motional heating, and motional dephasing on the decoded logical state. Moreover, we numerically analysed the fidelity of distributed logical Bell-pair fidelity. We find that one can achieve more than 95% of fidelity in the realistic situation. The protocol combines motional-control and photonic-networking operations that have already been demonstrated in trapped-ion experiments.