Extended Data Fig. 8: Entanglement heralding between two remote ¹⁷¹Yb ions using a two photon protocol.

a, Pulse sequence for remote entanglement generation. Each qubit is prepared in a superposition state, \(1/\sqrt{2}(| 0\rangle +| 1\rangle )\), and optically excited in two rounds, separated by a qubit π pulse. Photon detections in both rounds at stochastic times t₀ and t₁ (measured relative to the start of their respective heralding windows) carve out \(| 00\rangle \) and \(| 11\rangle \), respectively, thereby heralding entanglement. Subsequently, a dynamical decoupling sequence is applied with inter-pulse spacing 2τ_s = 5.8 μs. In cases where t₁ < t₀ optical and spin coherences are rephased for durations τ_h − t₀ + t₁ and t₀ − t₁, respectively, after an even number of dynamical decoupling periods. In cases where t₁ > t₀ optical and spin coherences are rephased for durations τ_h − t₁ + t₀ and t₁ − t₀, respectively, after an odd number of dynamical decoupling periods. Note that we also apply a Z rotation to Ion 2’s qubit by an angle \(\Delta {\widetilde{\omega }}_{12}({t}_{0}-{t}_{1})\) (not labelled). b, Entanglement rate and fidelity plotted against window size, W, where heralding events are accepted under the condition ∣t₁ − t₀∣ < W/2. The smallest window size, W = 200 ns, leads to a fidelity and rate of \({\mathcal{F}}=0.84\,\pm \,0.03\) and \({\mathcal{R}}=17\) mHz, respectively. The largest window size of W = 1.6 μs corresponds to a fidelity and rate of \({\mathcal{F}}=0.75\,\pm \,0.02\) and \({\mathcal{R}}=91\) mHz, respectively. c, Density matrix obtained via maximum likelihood tomography with a window size W = 600 ns (vertical dashed line in b) leading to a fidelity and rate of \({\mathcal{F}}=0.81\,\pm \,0.02\) and \({\mathcal{R}}=49\) mHz, respectively.