Symmetry: A Fundamental Resource for Quantum Coherence and Metrology.

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⟨G^{2}⟩. 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 the particle number for bosons. Our results establish the optimal use for metrology of arbitrarily non-Gaussian quantum correlations in a large variety of many-body systems.