Figure 1

(a) The SLOCC criterion divides the Hilbert space (the square) in such a way that every quantum state is in a well defined class (the lines). For four or more parties, the number of these SLOCC classes is infinite. However, they may be gathered into families (the colored areas) under certain rules, ideally with more physical associations than mathematical ones. Here, the condition is given by the minimal bond dimension of the matrix-product-state representation of quantum states, relating the MPS classes to the interaction length of parent Hamiltonians. (b) The proposed MPS classification enjoys a scalable nesting property in which the classes of an N-partite family can be mapped onto the classes of the corresponding (N + 1) case, generating a matryoushka structure. A detailed example is given for the symmetric subspace of arbitrary number of parties.