Due to my limited knowledge, when I read the paper by Mattia Zorzi. I am confused about its notation. Hope you can help me explain.
See Zorzi, Mattia; Ticozzi, Francesco; Ferrante, Augusto, Minimum relative entropy for quantum estimation: feasibility and general solution, IEEE Trans. Inf. Theory 60, No. 1, 357-367 (2014). ZBL1364.81074.,
- In equation 29, the authors wrote $\delta \mathcal{L}(\rho, \lambda; \delta \rho)$, I don’t know the meaning of the semicolon in brackets, i.e, $; \delta \rho$.
- In Equation 36, the authors wrote $\left\langle\lambda^{\perp}, \bar{f}\right\rangle$, where $\lambda=\left[\begin{array}{lll}\lambda_{1} & \ldots & \lambda_{m}\end{array}\right]^{T} \in \mathbb{R}^{m}$ and $\bar{f}=\left[\begin{array}{lll}\bar{f}_{1} & \ldots & \bar{f}_{m}\end{array}\right]^{T}$. I don’t know what the $\lambda^{\perp}$ means.
- In mathematics, what do $\operatorname{Range}$ and $\operatorname{range}$ stand for respectively. For example, the authors wrote "note that $\left\langle\lambda^{\perp}, \bar{f}\right\rangle=0$ for each $\lambda^{\perp} \in[\text { Range } L]^{\perp}$" and "Notice that in order to have bounded values of the entropy it is only necessary that range $(\rho) \subseteq \operatorname{range}(\tau)$".
Thanks for your help!
Best Answer
First thing to say, your question is slightly not about calculus of variations, because authors deal with finite-dimensional optimizations (minimizing functions with finite-dimensional argument), whereas calculus of variations is about infinite-dimensional arguments (for example, functions).
EDIT: added some detailed information to 3.