Criteria for SLOCC and LU Equivalence of Generic Multi-qudit States

TL;DR

This paper develops tensor-based criteria for determining SLOCC and LU equivalence of generic multi-qudit states, providing necessary and sufficient conditions using tensor decompositions for pure states and conditions for mixed states.

Contribution

It introduces a novel tensor decomposition approach to characterize SLOCC and LU equivalence, extending the analysis to both pure and mixed multi-qudit states.

Findings

Tensor-based criteria for pure states established

Necessary and sufficient conditions for LU equivalence in mixed states

Necessary conditions for SLOCC equivalence in mixed states

Abstract

In this paper, we study the stochastic local operation and classical communication (SLOCC) and local unitary (LU) equivalence for multi-qudit states by mode-$n$ matricization of the coefficient tensors. We establish a new scheme of using the CANDECOMP/PARAFAC (CP) decomposition of tensors to find necessary and sufficient conditions between the mode-$n$ unfolding and SLOCC\&LU equivalence for pure multi-qudit states. For multipartite mixed states, we present a necessary and sufficient condition for LU equivalence and necessary condition for SLOCC equivalence.