Special core tensors of multi-qubit states and the concurrency of three lines

Abstract

In this work, we propose a computationally simple approach to identify the local unitary (LU) entanglement classes of multi-qubit states by higher-order singular value decomposition (HOSVD). For multipartite states, HOSVD simultaneously diagonalizes their one-body reduced density matrices (RDM) by LU actions. Therefore, the zeros of the all-orthogonality conditions due to HOSVD, also known as the core tensors, are the pure-state representations of such simultaneously diagonalized one-body RDM for a given multipartite state. By using the concurrency of three lines, we simplified the calculations and coarse-grained the classification into a finite number of families of states based on the square of their first n-mode singular values, σ1(n)2 . These special core tensors are genuinely entangled by default. For three and four qubits, we identified two and four families of states respectively. A generalization of the algorithm to multi-qubit states is provided.