Quantum enhanced measurements aim at exploiting quantum mechanical effects for improving the sensitivity of measurements of classical parameters of a system. The most-well known examples are the improvement of the measurement of phase shifts in an interferometer by using squeezed light [1], or the idea of using NOON states for super-resolution [2]. In 1994 Braunstein and Caves developed a general theory of quantum enhanced measurements, based on a generalization to the quantum world of the celebrated Cramér-Rao bound in classical parameter estimation theory [3, 4]. Indeed, even classically the problem of the optimal strategy of determining the value of a parameter x that parametrizes a probability distribution, and the best achievable sensitivity, are of fundamental importance in many branches of science. In quantum mechanics, one would like to get the best possible estimate of a variable x that parametrizes a general quantum state ρ ( x ) , based on arbitrary measurements, repeated eventually N times in an always identically prepared state. In the above example of an interferometer, ρ ( x ) would be the state at the output of the interferometer, and x the phase shift we want to measure. The latter might itself depend on a number of interesting parameters, such as the length of an arm of the interferometer in the case of a Mach-Zehnder interferometer, the absorption index of a probe brought into one of the arms, or the magnetic field acting on a spin in the case of Ramsey interferometry. Braunstein and Caves showed [3] that the best achievable sensitivity for measuring x in the state ρ ( x ) is given by ( dx 2 ) Here, ds 2 = 4 d 2 $_{Bures}$( ρ ( x ) , ρ ( x ) + dρ ) is given by the so-called Bures distance, which will be defined precisely below, and dρ denotes the infinitesimal change of ρ ( x ) when x varies about its initial value for which we want to know the best achievable sensitivity. It is important to note that the bound (1) can be saturated in the limit of large N . Based on (1), a number of important results were obtained. Notably, it was shown that for an initial product state | ψ 〉 = ⊗ K i =1 | φ 〉 i of K sub-systems (i.e. K “quantum resources”), unitary evolution with a hamiltonian H ( x ) = x ∑ K i $_{=1}$h$_{i}$ , and arbitrary final projective mea- 〈 δx $^{2}$〉 1 / 2 ρ ( x ) ≥ 1 √$_{N}$ ds 2 1 / 2 ≡ δx$_{min}$ | ρ ( x ) . (1) 2