We analytically derive the lower error bound for the preparation of any maximally entangled state of two atoms involving Rydberg-state interactions. This fundamental bound represents the minimum achievable error
E≥(1+π/2)Γ/B due to spontaneous decay
Γ of the Rydberg states and their finite interaction strength
B, assuming that all other technical errors can be eliminated. 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.