Random symmetric states for robust quantum metrology

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We use powerful concentration of measure techniques to study how many states are useful for quantum metrology, i.e., give a precision in parameter estimation surpassing fundamental limits in the classical case. First, we show that random pure multiparticle states do not lead to quantum enhancement. Conversely, we prove that typical pure states on the symmetric (bosonic) subspace achieve Heisenberg scaling with probability approaching unity for any fixed Hamiltonian encoding. We generalize our results to random mixed states having the fixed spectrum and study the impact of particle losses. We find that surprisingly, in contrast to highly entangled GHZ state, random bosonic states typically exhibit Heinsenberg scaling even in the presence of finite particle losses. In addition, we prove that with random symmetric states a single fixed measurement implementable with a Mach-Zehnder interferometer and photon number detectors suffices to typically achieve the Heisenberg scaling for all values of the unknown phase parameter. Finally, we demonstrate that for two optical modes metrologically useful states can be efficiently prepared by means of random optical circuits generated from a set of gates consisting only of simple beam-splitters and a single non-linear (Kerr-like) transformation.