Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states

摘要

We construct coefficient matrices of size 2u0002 by 2n−u0002 associated with pure n-qubit states and prove theninvariance of the ranks of the coefficient matrices under stochastic local operations and classical communicationn(SLOCC). The ranks give rise to a simple way of partitioning pure n-qubit states into inequivalent familiesnand distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme vianthe ranks of coefficient matrices can be combined with other schemes to build a more refined classificationnscheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys.nRev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, andnas a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up tonpermutations of the four qubits. We also discuss the completeness of the classification of four qubits into ninenfamilies.