Microwave Analog Quantum Teleportation Protocol

• Microwave analog quantum teleportation is a deterministic protocol that transfers continuous-variable states by exploiting two-mode squeezed microwave entanglement.
• It utilizes flux-pumped JPAs, hybrid rings, and analog feedforward to perform Bell-type measurements and displacements, achieving fidelity beyond the no-cloning threshold.
• Experimental implementations confirm robust state transfer with high fidelity, paving the way for scalable, modular superconducting quantum processors and networks.

Microwave analog quantum teleportation protocols constitute a class of continuous-variable state transfer schemes wherein quantum information encoded in propagating microwave fields is deterministically transferred between spatially separated nodes by exploiting two-mode squeezed microwave entanglement and analog feedforward. Such protocols are central to the development of distributed superconducting quantum processors, quantum local area networks, microwave-to-optical quantum transduction, and various hybrid quantum networking applications. Key features include compatibility with superconducting circuit QED infrastructure, full determinism (no post-selection), and the ability to exceed the quantum no-cloning threshold for coherent state transfer.

1. Theoretical Framework: Quantum States, Squeezing, and Entanglement

Microwave analog quantum teleportation is executed using traveling microwave modes described by bosonic annihilation operators ($a$, $a^\dagger$) and their quadratures ($X=(a+a^\dagger)/\sqrt{2}$, $P=(a-a^\dagger)/(i\sqrt{2})$, $[X,P]=i/2$) (Fedorov et al., 2021). The entangled resource is a two-mode squeezed vacuum (TMSV) state $|\psi_\text{TMSV}\rangle = S_T(r_T)|0,0\rangle$ generated by a two-mode squeezing operation $$S_T(r_T) = \exp[r_T(a_1 a_2 - a_1^\dagger a_2^\dagger)]$$. In the ideal infinite-squeezing limit, the resource exhibits perfect Einstein-Podolsky-Rosen (EPR) correlations $(X_1-X_2)\to 0$, $(P_1+P_2)\to 0$. Finite squeezing is quantified in decibels: $S_T~[\text{dB}] \simeq -10\log_{10}(e^{-2r_T})$.

The input is typically a coherent microwave state $|\alpha\rangle = D(\alpha)|0\rangle$ (where $D(\alpha) = \exp[\alpha a^\dagger-\alpha^* a]$). The teleportation protocol operates optimally for input mean photon numbers up to $n_d = |\alpha|^2 \sim 1$.

2. Protocol Steps: Resource Preparation, Measurement, and Feedforward

Microwave analog teleportation proceeds in three deterministic stages (Fedorov et al., 2021):

1. Entanglement Resource Generation: Two flux-pumped Josephson Parametric Amplifiers (JPAs) generate single-mode squeezed microwaves; outputs are mixed on a 90° hybrid ring (microwave beam splitter) with phase difference $\pi/2$, yielding the TMSV (Fedorov et al., 2021). The actual squeezing at output ports accounts for insertion losses ($\sim 0.4$ dB) and stray thermal photons.
2. Bell-type Joint Measurement (Alice): The unknown input $|\alpha\rangle$ and Alice’s share of TMSV are combined on a second 90° hybrid ring, producing the necessary sum and difference quadratures. These are independently phase-sensitively amplified (measurement JPAs, degenerate gain $G$) and detected, yielding classical voltage signals representing $(x_m, p_m)$.
3. Analog Feedforward and Displacement (Bob): The measurement outcomes $(x_m, p_m)$ are routed via low-loss cryogenic cabling to Bob’s location and used as modulation amplitudes for classical microwave tones that are directed, via a weakly coupled directional coupler, onto Bob’s half of the entangled mode. Bob applies a displacement $D_B(\beta)$ with $\beta = g_x x_m + i g_p p_m$, where amplitude gains are typically optimized ($g_x = g_p = 1$ in the ideal case).

Every instance of the protocol yields an output; no post-selection or heralding is required, in contrast to photon-counting optical schemes (Fedorov et al., 2021).

3. Experimental Realization: Circuit Components and Noise Sources

Implementations employ standard circuit QED elements (Fedorov et al., 2021):

• Entanglement JPAs: $\lambda/4$ aluminum resonators terminated by SQUID arrays.
• Measurement JPAs: Degenerate parametric amplifiers (phase-sensitive mode, gain up to $G=21$–$23$ dB).
• Hybrid Rings and Couplers: Microwave beam splitters (insertion loss $\sim$0.4 dB).
• Superconducting Coaxial Cables: Ultra-low loss ($<0.001$ dB/m at $5$ GHz); typical link length $d=42$ cm.
• Amplification and Readout: Microwave isolators, HEMT amps, FPGA-based digitization/processing (measurement bandwidth up to 400 kHz).

Dominant sources of infidelity include insertion losses ($L \sim 2$ dB), added JPA noise photons, and bandwidth/compression limits (JPA $1$ dB compression at $-130$ dBm).

4. Fidelity Analysis and Quantum Advantage

The fidelity for teleporting an input coherent state is defined as $$F(\alpha) = \langle \alpha | \rho_\text{out} | \alpha \rangle$$. Classical "measure-and-prepare" strategies cannot surpass the no-cloning threshold $F_\text{nc} = 2/3$, and any quantum protocol exceeding this value demonstrates genuine quantum teleportation (Fedorov et al., 2021). The reported experiment achieves $F = 0.689 \pm 0.004$ for $n_d \leq 1.1$ (measured against the benchmark $F_\text{nc}=2/3$) with $6$ dB two-mode squeezing and $21$ dB measurement gain.

Scaling fidelity with squeezing and gain, the protocol exhibits an optimum measurement gain ($$G_\text{opt} \simeq -\beta + 6\,\text{dB} + L \simeq 21$$–$23$ dB$)$ for a fixed squeezing $S$, and the maximal fidelity increases with larger $S$. Achieving $F \to 0.8$ necessitates $S \gtrsim 8$ dB and $G \gtrsim 25$ dB. With finite squeezing or nonideal gain, the output becomes thermalized around the target displacement (Fedorov et al., 2021).

5. Deterministic Microwave CV Teleportation: Protocol Variants and Noise Models

Microwave analog quantum teleportation is inherently deterministic due to the use of analog feedforward, as opposed to optical protocols that require probabilistic Bell-state measurements or photon detection (Fedorov et al., 2021). Classical communication of continuous measurement outcomes $(x_m, p_m)$ within the cryostat ensures real-time deterministic completion.

Feedforward transmission can be affected by analog noise and loss in the classical channel, but appropriate gain calibration can fully correct for these imperfections without impacting security thresholds for finite-energy codebooks (Yam et al., 29 Dec 2025). The security of the protocol against public-channel eavesdroppers is governed by mutual information and Holevo bounds rather than the conventional no-cloning threshold.

6. Applications: Modular Quantum Networks and Prospects for Scaling

Microwave analog quantum teleportation protocols are compatible with modular superconducting processor platforms, circuit QED elements, and can be integrated into quantum gate architectures for distributed quantum computing (Fedorov et al., 2021). By extending the protocol to more complex codewords (e.g., GKP or cat codes), logical qubits may be teleported over microwave links.

Scaling to longer distances in quantum local area networks will require lower-loss guides (e.g., transmission loss $L \ll 0.1$ dB/m) or quantum-limited transduction mechanisms. Experimental demonstrations confirm robust operation at the scale of $42$ cm, with technological improvements in squeezing, JPA design, and cable loss projected to push fidelities toward unity and enable practical short-range quantum networking (Fedorov et al., 2021).

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Key Paper References

• Fedorov et al., “Experimental quantum teleportation of propagating microwaves” (Fedorov et al., 2021)
• Wu et al., “Practical quantum teleportation with finite-energy codebooks” (Yam et al., 29 Dec 2025)