I understand that one method to derive an MPS representation of a quantum state involves applying the Schmidt decomposition $ N−1$ times. While I'm familiar with the diagrammatic notation, I wanted to understand the math behind by working through a simple tripartite state to get its MPS form.
Given a general tripartite state:
$$|\psi\rangle = \sum_{i,j,k} \psi_{i,j,k} |i,j,k\rangle.$$
We follow the steps:
Step 1: Partition the State
Partition the state into systems labeled by $\color{blue}{i}$ and $\color{red}{jk}$ yielding: $$|\psi\rangle = \sum_{i,j,k} \psi_{\color{blue}{i},\color{red}{jk}} |i,j,k\rangle.$$
Step 2: First Schmidt Decomposition
Now, we'll apply the Schmidt decomposition to this partition:
$$|\psi\rangle = \sum_{\alpha}\sum_{i,j,k} U_{i,\alpha} \Lambda_{\alpha,\alpha} V^*_{jk,\alpha} |i,j,k\rangle = \sum_{\alpha} \Lambda_{\alpha,\alpha} \color{blue}{\left(\sum_{i} U_{i,\alpha} |i\rangle \right)}\otimes \color{red}{\left(\sum_{j,k}V^*_{jk,\alpha} |j,k\rangle \right)}.$$
Question 1. I don't understand the term red term. Specifically, the matrix element $V_{jk,\alpha}$. Does this suggest that for each column $\alpha$, there is a corresponding $j \times k$ matrix? My question arises because the next step involves applying a Schmidt decomposition to the term highlighted in red. While I can rewrite this term as $V^{(\alpha)}_{j,k}$, I'm uncertain about its mathematical implications. What should I do? I read about "re-shaping the matrix", but I don't understand either what does that mean and how the dependence of alpha is carried by doing so.
