Quantum R\'enyi entropy by optimal thermodynamic integration paths

Despite being a well-established operational approach to quantify entanglement, R\'enyi entropy calculations have been plagued by their computational complexity. We introduce here a theoretical framework based on an optimal thermodynamic integration scheme, where the R\'enyi entropy can be efficiently evaluated using regularizing paths. This approach avoids slowly convergent fluctuating contributions and leads to low-variance estimates. In this way, large system sizes and high levels of entanglement in model or first-principles Hamiltonians are within our reach. We demonstrate it in the one-dimensional quantum Ising model and perform the evaluation of entanglement entropy in the formic acid dimer, by discovering that its two shared protons are entangled even above room temperature.