This talk will consist of two parts. In the first part,

I will present the irreducible form of a Matrix Product States (MPS),

which is a generalization of the canonical form of an MPS in the

sense that it is also defined for states with periodicity. I will

then present a fundamental theorem for MPS in irreducible form,

namely one that specifies how two tensors in irreducible form are

related if they give rise to the same MPS. Finally, I will present

two applications of this result: an equivalence between the

refinement properties of a state and the divisibility properties

of its transfer matrix, and a more general characterisation of

tensors that give rise to matrix product states with symmetries.

In the second part, I will present a study of continuum limits

of MPS, where we show that an MPS has a continuum limit (for a

proper definition thereof) if and only if its transfer matrix

is an infinitely divisible channel. We also consider continuum

limits after a finite number of coarse graining steps, and

characterize it in terms of the divisibility properties of the

transfer matrix. I will present several examples of states with

and without the two kinds of continuum limits.

Joint work with I. Cirac, D. Perez-Garcia and N. Schuch.

Based on arXiv:1708.00880 and arxiv:1708.00029.