Achieving Heisenberg scaling with maximally entangled states: An analytic upper bound for the attainable root-mean-square error

In this paper we explore the possibility of performing Heisenberg limited quantum metrology of a phase, without any prior, by employing only maximally entangled states. Starting from the estimator introduced by Higgins et al. [New J. Phys. 11, 073023 (2009)NJOPFM1367-263010.1088/1367-2630/11/7/073023], the main result of this paper is to produce an analytical upper bound on the associated mean-squared error which is monotonically decreasing as a function of the square of the number of quantum probes used in the process. The analyzed protocol is nonadaptive and requires in principle (for distinguishable probes) only separable measurements. We explore also metrology in the presence of a limitation on the entanglement size and in the presence of loss.