Surpassing the Global Heisenberg Limit Using a High-effciency Quantum Switch

Indefinite causal orders have been shown to enable a precision of inverse square N in quantum parameter estimation, where N is the number of independent processes probed in an experiment. This surpasses the widely accepted ultimate quantum precision of the Heisenberg limit, 1/N. While a recent laboratory demonstration highlighted this phenomenon, its validity relies on postselection for it only accounted for a subset of the resources used. Achieving a true violation of the Heisenberg limit-considering photon loss, detection ineffciency, and other imperfections-remains an open challenge. Here, we present an ultrahigh-effciency quantum switch to estimate the geometric phase associated with a pair of conjugate position and momentum displacements embedded in a superposition of causal orders. Our experimental data demonstrate precision surpassing the global Heisenberg limit without artificially correcting for losses or imperfections. This work paves the way for quantum metrology advantages under more general and realistic constraints.