Certification Time for Weak-Force-Mediated Entanglement between Levitated Particles

Abstract

Certifying motional entanglement generated by weak forces between levitated nanoparticles requires distinguishing entangled from separable states using finite, noisy measurement records. We show that the number of measurements required to certify Gaussian entanglement can vary among equally entangled states, and quantify this variation. We derive an attainable lower bound showing that, near the separability threshold, the required number of measurements diverges inversely with the square of the distance from this threshold. We translate the required number of measurements into an experimental certification time, comparing two protocols under comparable experimental conditions: release--recapture, in which particles evolve freely before measurement, and continuous monitoring, in which the steady state is measured continuously. We find that release--recapture enables entanglement certification at interaction strengths several orders of magnitude weaker. Moreover, minimizing certification time can favor states with less entanglement, revealing a trade-off between entanglement generation and its certification. These results highlight the role of certification in the design and optimization of ongoing efforts to probe weak-force-mediated entanglement between levitated nanoparticles.

The paper asks how long it takes to certify weak-force-generated motional entanglement between levitated particles, rather than merely generate it. It considers unconditional two-mode Gaussian states reconstructed from noisy quadrature measurements, using the smallest partially transposed symplectic eigenvalue $\tilde\nu_-$ as the entanglement witness. Gaussian covariance-estimator statistics and the delta method give an asymptotic measurement requirement $N_\star\simeq[n_\sigma\alpha/(1-\tilde\nu_-)]^2$. For isotropic added measurement noise $\mu I$, it derives the attainable bound $\alpha\geq\tilde\nu_-+\mu$, saturated by symmetric thermal two-mode squeezed states. Thus the minimum sampling cost diverges as $(1-\tilde\nu_-)^{-2}$ near separability. Local squeezing preserves entanglement but can make the sampling cost arbitrarily larger under the fixed readout. The framework is applied to release–recapture dynamics with momentum diffusion and to continuously monitored steady states with specified linear–quadratic–Gaussian feedback. At fixed diffusion-to-coupling ratio and interaction phase, release–recapture gives $N_\star\propto g^{-1}$ and, when release dominates each cycle, certification time $T_\star\propto g^{-3/2}$. Optimizing release duration for certification rather than maximum entanglement yields severalfold time reductions in the examples. Under the compared conditions, release–recapture certifies interactions several orders of magnitude weaker than continuous monitoring. The analysis is asymptotic in sample size, assumes Gaussian dynamics and perfectly calibrated measurement noise, and neglects calibration uncertainty. The quadratic interaction requires relative displacements small compared with particle separation. The protocol comparison uses a particular feedback implementation; available release/free-fall time can further restrict experimental operation.