# How to find the $A_i$ in the matrix product state representation?

From what I understand, MPS is just a simpler way to write out a state, compared to the density matrix. But how do I get those $$A_i$$ matrices? From all the examples I read, people just somehow "have" the matrices in their pocket already. Is there a way to generate them from some random state, say $$|\Psi\rangle = \alpha|\uparrow\rangle^{\otimes N}+\beta|\downarrow\rangle^{\otimes N}$$, or some random density matrix? Feel free to set $$N$$ to small numbers if it becomes too complicated, I just really want to see an example of how this works in general, not just a GHZ state! Thanks a lot!

• The two best sources of information I could find were : the wikipedia page and this arxiv paper May 21 at 9:45
• Well, I read both of them. The Wikipedia page is exactly the one without any derivation for $A_i$, and the paper doesn't really have any kind of example. May 21 at 10:18
• I couldn't find anything either, it only says that schmidt decomposition is required, when someone aswers, we should update the wikipedia page ! May 21 at 11:30
• The explanation in Vidal's paper is much more explicit arxiv.org/abs/quant-ph/0301063 May 21 at 15:49
• See this answer from physics StackExchange: physics.stackexchange.com/a/565749/252841 May 22 at 6:37

On the other hand, this description is usually of interest if the resulting MPS description has much less parameters than the $$2^N$$ parameters needed to describe a general quantum state. This is, e.g., the case for the example you give.