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 $E\ge (1+\pi \mathrm{/}2)\mathrm{\Gamma }\mathrm{/}B$ due to spontaneous decay $\mathrm{\Gamma }$ 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 $1\mathrm{\%}$ above the derived fundamental bound.