  We investigate how the Hilbert-Schmidt speed (HSS), a special type of quantum
statistical speed, can be exploited as a powerful and easily computable tool
for quantum phase estimation in a $n$-qubit system. We find that, when both the
HSS and quantum Fisher information (QFI) are computed with respect to the phase
parameter encoded into the initial state of the $n$-qubit register, the zeros
of the HSS dynamics are essentially the same as those of the QFI dynamics.
Moreover, the positivity (negativity) of the time-derivative of the HSS exactly
coincides with the positivity (negativity) of the time-derivative of the QFI.
Our results also provide strong evidence for contractivity of the HSS under
completely positive and trace preserving maps in high-dimensional systems, as
predicted in previous studies.

Quantum Physics