Fundamental bound on entanglement generation between interacting Rydberg atoms

We analytically derive the fundamental lower bound for the preparation fidelity of a maximally-entangled (Bell) state of two atoms involving Rydberg-state interactions. This bound represents the minimum achievable error $E \geq ( 1 + π/2 ) Γ/B$ due to spontaneous decay $Γ$ of the Rydberg states and their finite interaction strength $B$. Using quantum optimal control methods, we identify laser pulses for preparing a maximally-entangled state of a pair of atomic qubits with an error only $1%$ above the derived fundamental bound.

💡 Research Summary

In this work the authors address a central limitation in neutral‑atom quantum processors that rely on Rydberg‑state interactions: the unavoidable error introduced by spontaneous decay of the Rydberg levels during the creation of maximally entangled two‑qubit states. By considering two atoms, each possessing at least two internal electronic states (one of which is a Rydberg state |r⟩), they write the total Hamiltonian as the sum of a dispersive interaction term H_int = B|rr⟩⟨rr| and a generic local laser‑driving term H_loc. The decay of the Rydberg level with rate Γ is incorporated via a non‑Hermitian contribution –iΓΠ_r/2, where Π_r projects onto the Rydberg subspace.

The goal is to transform the product ground state |ψ_in⟩ = |gg⟩ into a Bell state |ψ_f⟩ = (|gg⟩ + |gr⟩ + |rg⟩ – |rr⟩)/2 with the highest possible fidelity while minimizing the accumulated decay error E = Γ T_r, where T_r = ∫_0^T P_r(t) dt and P_r(t) is the instantaneous Rydberg population. To quantify entanglement growth they introduce the Schmidt decomposition |ψ(t)⟩ = ∑_i c_i(t)|u_i(t)⟩|v_i(t)⟩ and define the “min‑entropy” S = −log₂(c_max²), with c_max the largest Schmidt coefficient. S evolves from 0 (product state) to 1 (maximally entangled two‑qubit state).

Using the Schrödinger equation they show that the time derivative of S depends only on the interaction Hamiltonian, yielding ˙S = (2/ln 2) Im⟨u₁v₁|H_int|ψ⟩/c_max. Applying the co‑area formula to the integral over time, they obtain a lower bound for the average Rydberg dwell time:

 T_r ≥ ∫₀¹ G(s) ds, with G(s) = min_{S(ψ)=s}