Abstract
We introduce a new paradigm for the preparation of deeply entangled states useful for quantum metrology. We show that when the quantum state is an eigenstate of an operator A, observables G which are completely off-diagonal with respect to A have purely quantum fluctuations, as quantified by the quantum Fisher information, namely F_Q(G)=4 G² . This property holds regardless of the purity of the quantum state, and it implies that off-diagonal fluctuations represent a metrological resource for phase estimation. In particular, for many-body systems such as quantum spin ensembles or bosonic gases, the presence of off-diagonal long-range order (for a spin observable, or for bosonic operators) directly translates into a metrological resource, provided that the system remains in a well-defined symmetry sector. The latter is defined e.g. by one component of the collective spin or by its parity in spin systems; and by a particle-number sector for bosons. Our results establish the optimal use for metrology of arbitrarily non-Gaussian quantum correlations in a large variety of many-body systems.