Stochastic local operations and classical communication (SLOCC) and local unitary operations (LU) classifications of n qubits via ranks and singular values of the spin-flipping matrices

We construct \(\ell \)-spin-flipping matrices from the coefficient matrices of pure states of n qubits and show that the \(\ell \)-spin-flipping matrices are congruent and unitary congruent whenever two pure states of n qubits are SLOCC and LU equivalent, respectively. The congruence implies the invariance of ranks of the \(\ell \)-spin-flipping matrices under SLOCC and then permits a reduction of SLOCC classification of n qubits to calculation of ranks of the \(\ell \)-spin-flipping matrices. The unitary congruence implies the invariance of singular values of the \(\ell \)-spin-flipping matrices under LU and then permits a reduction of LU classification of n qubits to calculation of singular values of the \(\ell \)-spin-flipping matrices. Furthermore, we show that the invariance of singular values of the \(\ell \)-spin-flipping matrices \(\Omega _{1}^{(n)}\) implies the invariance of the concurrence for even n qubits and the invariance of the n-tangle for odd n qubits. Thus, the concurrence and the n-tangle can be used for LU classification and computing the concurrence and the n-tangle only performs additions and multiplications of coefficients of states.