Following this paper, the quantum Fisher information matrix (QFIM) - $\mathcal{F}$ can be calculated as:
$\mathcal{F}_{i, j}(\theta)=4 \operatorname{Re}\left[\left\langle\partial_{i} \psi(\boldsymbol{\theta}) \mid \partial_{j} \psi(\boldsymbol{\theta})\right\rangle-\left\langle\partial_{i} \psi(\boldsymbol{\theta}) \mid \psi(\boldsymbol{\theta})\right\rangle\left\langle\psi(\boldsymbol{\theta}) \mid \partial_{j} \psi(\boldsymbol{\theta})\right\rangle\right]$
$|\psi(\theta)\rangle$ is the current quantum state and $\theta$ is the $N$-dimensional complex vector, that means $\mathcal{F}$ is the $N\times N$ matrix.
The thing that I confused is $\partial_{k} \psi(\boldsymbol{\theta})$ is a scalar so how to calculate $\langle\partial_{i} \psi(\boldsymbol{\theta}) \mid \psi(\boldsymbol{\theta}) \rangle$ and its dagger?