Quantum reverse engineering and reference-frame alignment without nonlocal correlations

The estimation of unknown qubit elementary gates and the alignment of reference frames are formally the same problem. Using quantum states made out of $N$ qubits, we show that the theoretical precision limit for both problems, which behaves as $1∕N^2$, can be asymptotically attained with a covariant protocol that exploits the quantum correlation of internal degrees of freedom instead of the more fragile entanglement between distant parties. This cuts by half the number of qubits needed to achieve the precision of the dense covariant coding protocol.