Quantum Cramér-Rao bound based on the Bures distance

Abstract

The quantum Fisher information can be different with the Bures metric and becomes discontinuous at the parameter points where the rank of the parametric density operator changes. We show that the invalidation of the QCRB at such parameter points is due to the fact that the symmetric logarithmic derivative operator tends to be unbounded when the value of parameter approaches these singular parameter points. We give an alternative derivation of the QCRB through the Bures distance and show that this form of the QCRB still holds even when the parametric density operator changes its rank.

Quantum Polarimetry

“…Many other details about quantum estimation can be found in the above-mentioned reviews and we, further, draw attention to recent work investigating situations in which the QFIM in Eq. ( 129) cannot be inverted or changes discontinuously (Goldberg and James, 2018;Goldberg et al, 2021e;Seveso et al, 2019;Šafránek, 2017;Ye and Lu, 2021;Zhou and Jiang, 2019).…”