Berry curvature as a lower bound for multiparameter estimation

Quantum Fisher information (QFI) is a key concept in quantum state estimation, and Berry curvature (BC) is another basic concept to describe geometric properties of quantum states. In this paper, we consider pure states undergoing unitary parametrization processes and show that the BC serves as a lower bound for the product of QFIs corresponding to two different parameters through the Heisenberg uncertainty relation. This relation between QFI and BC implies that the estimation precisions of two different parameters are mutually restrictive due to finite BC, and the notion of QFI squeezing is introduced. A scenario of general su(2) parametrization is considered in detail to verify the relation between the QFI and BC.

Figure 1

Plot of four times the square of BC (dashed red line) together with the product of QFIs (solid blue line) of $\theta$ and $\varphi$, with $j=1$. It is to be noted that both QFI and BC are dimensionless due to the dimensionless $\theta$ and $\varphi$. The parameters are (a) $\theta =\varphi =\pi /4$ and the square of curvature oscillates along with the product of the QFIs; (b) $\theta =\pi /2$ and $\varphi =\pi /4$ and two lines almost coincide while actually there exists a tiny difference between them; (c) for $t=3$ and $\varphi =\pi /4$ the lower bound becomes tight near the point $\theta =\pi /2$; and (d) for $t=3$ and $\theta =\pi /3$ the lower bound becomes tight around the point $\varphi =\pi /2$.

Figure 2

Plot of the QFI squeezing parameters, ${\xi }_{\theta }$ (brown line) and ${\xi }_{\varphi }$ (dashed green line), during the time evolution with parameter setting $\theta =\pi /2$ and $\varphi =\pi /4$ (with $j=1$); the squeezing parameters are dimensionless since QFI and BC are dimensionless. As shown in Eq. (26) that a less-than-one squeezing parameter guarantees its dual squeezing parameter to be larger than one, it is clear to see that when one squeezing parameter is below 1 (a relative smaller QFI) another squeezing parameter rises above 1 (a relative larger QFI), but not vice versa.