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!

Crosspost with: https://physics.stackexchange.com/questions/638206/how-to-translate-from-a-state-density-matrix-formalism-to-matrix-product-state-r

  • $\begingroup$ The two best sources of information I could find were : the wikipedia page and this arxiv paper $\endgroup$ Commented May 21, 2021 at 9:45
  • $\begingroup$ 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. $\endgroup$ Commented May 21, 2021 at 10:18
  • $\begingroup$ I couldn't find anything either, it only says that schmidt decomposition is required, when someone aswers, we should update the wikipedia page ! $\endgroup$ Commented May 21, 2021 at 11:30
  • $\begingroup$ The explanation in Vidal's paper is much more explicit arxiv.org/abs/quant-ph/0301063 $\endgroup$ Commented May 21, 2021 at 15:49
  • $\begingroup$ See this answer from physics StackExchange: physics.stackexchange.com/a/565749/252841 $\endgroup$ Commented May 22, 2021 at 6:37

1 Answer 1


Any state can be written as a matrix product state. There are systematic procedures to construct such a description, based on sequential SVDs, see e.g. Section 4.1.3 of this review.

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.


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