δx$_{min}$ | L x [ | ψ i 〉〈 ψ i | $_{]}$. Plugging this upper bound on d 2 Bures back into (1) and taking the square root immediately shows the theorem. Since there are many ways of mixing a given density matrix from pure states, the usefulness of the theorem for obtaining bounds on the best sensitivity achievable may depend on the chosen decomposition of ρ$_{0}$ . For example, one may start with a pure state that is entangled over all subsystems, and admix to it a separable state (see e.g.[7]). Then the lower bound on δx$_{min}$ remains given by the entangled state. However, for a sufficiently large admixture of the separable state, the initially entangled state might become at least partially separable, and a new decomposition of ρ ( x ) in terms of pure states can be found, which raises the lower bound on δx$_{min}$ . The most straightforward and useful application of the theorem arises in the case of an initially fully separable state ρ$_{0}$ . In this case the theorem immediately shows that one cannot achieve a better sensitivity than with the initial pure product states from which it is mixed, i.e. in particular a scaling as 1 $_{/}$√$_{K}$ with the number of quantum resources K for the example mentioned in the introduction. This generalizes the findings of [5] for the CC and CQ scenarios (classical initial state, taken as a pure product state in [5], and factorizing or entangling measurement) to mixed separable states. Very recently a preprint appeared which came to the same conclusion using a different method, and in a less general setting [8]. [1] C. M. Caves, Phys. Rev. D 23 , 1693 (1981). [2] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Physical Review Letters 85 , 2733 (2000). [3] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72 , 3439 (1994). [4] H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, NJ, 1946). [5] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96 , 010401 (2006). [6] J. A. Miszczak, Z. Puchała, P. Horodecki, A. Uhlmann, and K.Życzkowski, Quantum Informa- tion and Computation 9 , 0103 (2009). 5