Consider a pure state $\boldsymbol{\eta} \in \mathcal{H}_{AB}$. There exist orthonormal sets $\{\alpha_1, \alpha_2 \dots \alpha_i\} \subset \mathcal{H}_A$ and $\{\beta_1, \beta_2 \dots \beta_i\} \subset \mathcal{H}_B$, and real numbers $\lambda_k > 0$ such that \begin{equation*} \boldsymbol{\eta} = \sum_{i=1}^d \lambda_i \alpha_i \otimes \beta_i \end{equation*} My question is if it is it possible (if so, how?) to find $d$ without using the decomposition above.
What I have done so far is that I have set $\text{dim}\mathcal{H}_A=m$ and $\text{dim}\mathcal{H}_B = n$. This means that $d \leq \min(m,n)$. Besides this, I do not know what to do. I know about a theorem called Caratheodory's theorem, but I am not sure if it will help me here. Can I use any of this to show that $d$ only depends on $\boldsymbol{\eta}$? Thanks!