The product structure of MPS-under-permutations

TL;DR

This paper demonstrates that translationally-invariant matrix product states with permutational symmetry are trivial, implying simpler ansätze may suffice for such symmetric systems.

Contribution

It proves that TI MPS with permutational symmetry are trivial, extending the result to non-TI MPS and relevant examples like W and Dicke states.

Findings

TI MPS with permutational symmetry are either product states or superpositions.

Results apply approximately to non-TI MPS and specific states like W and Dicke states.

Suggests using simpler ansätze than tensor networks for permutation-invariant systems.

Abstract

Tensor network methods have proved to be highly effective in addressing a wide variety of physical scenarios, including those lacking an intrinsic one-dimensional geometry. In such contexts, it is possible for the problem to exhibit a weak form of permutational symmetry, in the sense that entanglement behaves similarly across any arbitrary bipartition. In this paper, we show that translationally-invariant (TI) matrix product states (MPS) with this property are trivial, meaning that they are either product states or superpositions of a few of them. The results also apply to non-TI generic MPS, as well as further relevant examples of MPS including the W state and the Dicke states in an approximate sense. Our findings motivate the usage of ans\"atze simpler than tensor networks in systems whose structure is invariant under permutations.