Hilbert-Schmidt speed as an efficient tool in quantum metrology

TL;DR

This paper demonstrates that Hilbert-Schmidt speed (HSS) is an effective, easily computable tool for quantum phase estimation, closely mirroring quantum Fisher information (QFI) dynamics in multi-qubit systems.

Contribution

It shows that HSS can serve as a practical alternative to QFI for quantum metrology, with similar dynamical behavior and contractivity properties in high-dimensional systems.

Findings

HSS zeros align with QFI zeros during phase estimation.

Time-derivative signs of HSS match those of QFI, indicating similar dynamical features.

HSS exhibits contractivity under quantum channels in high-dimensional systems.

Abstract

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.