Loss-tolerant parallelized Bell-state generation with a hybrid cat qudit

TL;DR

This work tackles scalable entanglement distribution by proposing loss-tolerant, parallelized Bell-state generation between distant qubit registers. It introduces two strategies: a phase-qudit encoding and a loss-detecting hybrid cat-qudit encoding, both driven by a phase-encoded coherent pulse and heralded by measurement outcomes. The cat-qudit approach leverages photon-number parity via XX parity checks to detect and correct single-photon losses, yielding favorable fidelity scaling $F_{\text{cat}}(n_\ell) = 1 - O(n_\ell^2)$ relative to phase encoding, and it provides analytic optimization for photon budgets using Lambert-W functions. The analysis shows that, for modest loss and small $N$, cat encoding substantially lowers channel-quality requirements, offering a practical path toward higher-rate quantum networks on both optical and circuit-QED platforms.

Abstract

Having multiple Bell pairs shared by distant quantum registers provides a key resource for both quantum networks and distributed quantum computing. In this paper, we present a protocol for parallelized Bell-pair generation that uses the phase of a coherent light pulse to encode a qudit, enabling the simultaneous generation of multiple Bell pairs. By encoding a qudit in a basis of light-matter Schrödinger's cat states, the loss of a photon in transit can be detected through an $XX$ parity syndrome, allowing the backaction due to the lost photon to be deterministically corrected through single-qubit rotations. The protocol presented here is compatible with existing technologies in both optical and microwave (circuit QED) architectures, supporting near-term implementation across diverse quantum platforms.

Figures (3)

• Figure 1: Schematic of the setup: Alice and Bob each possess a register of $N$ qubits numbered $0,\dots,N-1$ (here, $N=2$), with each qubit coupled to a single quantized cavity mode. In the loss-tolerant version of the protocol, Alice and Bob need one additional ancilla each, here labeled A and B, respectively (in gray). A coherent light pulse prepared by Alice becomes entangled first with Alice's qubits then with Bob's qubits through a series of qubit-state-conditioned phase shifts imparted upon reflection from each cavity in turn. A heterodyne measurement of the light pulse, together with a circuit applied locally on Bob's register, can then be used to produce $N$ Bell pairs shared by Alice and Bob. The loss of a photon in transit can be detected by also entangling the light pulse with the ancillas A and B. In this case, a measurement of the ancilla-qubit $XX$ parity will reveal whether a photon was lost. Conditioned on an eigenvalue $XX=-1$, Alice can correct the backaction due to the loss by applying single-qubit rotations to her register. The ancillas could either be coupled to their own separate cavities or to the same cavities as one of Alice's and Bob's register qubits (as pictured here).
• Figure 2: Phase-space representation of the entangling operation for a phase-encoded qudit and $N=3$: For each of the $2^N=8$ basis states of Alice's qubits, an initial coherent state $\ket{\alpha}$ undergoes a phase-space rotation by an amount $m\varphi$, where $\varphi=2\pi/2^{N}$ and $m=\sum_{j=0}^2 s_i 2^i$ is the base-10 representation of the computational basis state $\ket{s_2 s_1 s_0}$ ($s_i=0,1$).
• Figure 3: Infidelity $\epsilon_\zeta=\epsilon_\zeta(N,\eta,n_\zeta)$ with $N=2$ as a function of the photon-loss probability $1-\eta$, where here, $n_\zeta$ is an optimized average photon number defined via Eq. \ref{['nopt']}. The black triangles (squares) correspond to a numerical evaluation of Eq. \ref{['nopt']} for $\zeta=\mathrm{cat}$ ($\zeta=\mathrm{phase}$), while the dashed (dot-dashed) line corresponds to the approximate closed-form solution $\Tilde{\epsilon}_\mathrm{cat}$ ($\Tilde{\epsilon}_\mathrm{phase}$) given in Eqs. \ref{['epsilon-opt']}-\ref{['approximate-infidelity']}, which is valid only for $\Lambda_{N,\eta}\propto 1-\eta< 1$.