Lower bounds on entanglement entropy without twin copy

We discuss the possibility of estimating experimentally the von Neumann entanglement entropy $S_{A}^{vN}$ of a symmetric bi-partite quantum system $AB$ by using the basic measurement counts (bitstrings) for a $single$ copy of a prepared state. Using exact diagonalization and analog simulations performed with the publicly available QuEra facilities for chains and ladders of Rydberg atoms, we calculate the Shannon entropy $S_{AB}^X$ associated with the bitstrings of adiabatically prepared ground states and the reduced entropies $S_A^X$ and $S_B^X$ obtained from the marginal probabilities in $A$ and $B$. We then calculate the classical mutual information $I^X_{AB}=S_A^X+S_B^X-S_{AB}^X$, which is a lower bound on $S_{A}^{vN}$. We show that for a broad range of lattice spacing and detuning, $I^X_{AB}$ is typically 20 percent below $S_{A}^{vN}$ in regions where $S_{A}^{vN}$ is large and a less close bound in regions where $S_{A}^{vN}$ is low. We argue that this use of the easily available bitstrings provides a robust and efficient way to explore empirically the phase diagram of qubit-based quantum simulators and identify critical regions.