  The geometry of quantum states provides a unifying framework for estimation
processes based on quantum probes, and it allows to derive the ultimate bounds
of the achievable precision. We show a relation between the statistical
distance between infinitesimally close quantum states and the second order
variation of the coherence of the optimal measurement basis with respect to the
state of the probe. In Quantum Phase Estimation protocols, this leads to
identify coherence as the relevant resource that one has to engineer and
control to optimize the estimation precision. Furthermore, the main object of
the theory i.e., the Symmetric Logarithmic Derivative, in many cases allows to
identify a proper factorization of the whole Hilbert space in two subsystems.
The factorization allows: to discuss the role of coherence vs correlations in
estimation protocols; to show how certain estimation processes can be
completely or effectively described within a single-qubit subsystem; and to
derive lower bounds for the scaling of the estimation precision with the number
of probes used. We illustrate how the framework works for both noiseless and
noisy estimation procedures, in particular those based on multi-qubit
GHZ-states. Finally we succinctly analyze estimation protocols based on
zero-temperature critical behaviour. We identify the coherence that is at the
heart of their efficiency, and we show how it exhibits the non-analyticities
and scaling behaviour proper of a large class of quantum phase transitions.

Quantum Physics