Classification of general n-qubit states under stochastic local operations and classical communication in terms of the rank of coefficient matrix

TL;DR

This paper introduces a method to classify n-qubit states under SLOCC by analyzing the rank of their coefficient matrices, simplifying entanglement classification and revealing inequivalence among Dicke states.

Contribution

It provides a new classification scheme based on coefficient matrix rank that applies to general n-qubit states and identifies inequivalence among symmetric Dicke states.

Findings

Rank of the coefficient matrix is invariant under SLOCC.

States with different ranks belong to different entanglement classes.

Dicke states with different l are inequivalent under SLOCC.

Abstract

We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |l,n> with l=1, ..., [n/2] are inequivalent under SLOCC.