We propose a method of classifying $n$-qubit states into stochastic local operations and classical communication inequivalent families in terms of the rank of the square matrix $C{\left(i{\sigma }_y\right)}^{\otimes k}{C}^T$, where $C$ is the rectangular coefficient matrix of the state and ${\sigma }_y$ is the Pauli operator. The rank of the square matrix $C{\left(i{\sigma }_y\right)}^{\otimes k}{C}^T$ is capable of distinguishing between $n$-qubit Greenberger-Horne-Zeilinger and $W$ states. The determinant of the matrix gives rise to a family of polynomial invariants for $n$ qubits which include as special cases well-known polynomial invariants in the literature. In addition, explicit expressions can be given for these polynomial invariants and this allows us to investigate the properties of entanglement measures built upon the absolute values of polynomial invariants for product states.

• Received 2 March 2014

DOI:https://doi.org/10.1103/PhysRevA.91.012302