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

Abstract

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.

Appendix A: Some expressions

$$\begin{aligned} e_{12}= & {} \sum _{i=0}^{2^{n-1}-1}(-1)^{N(i)}a_{i}a_{2^{n}-1-i},\\ e_{11}= & {} 2\sum _{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{i}a_{2^{n-1}-1-i},\\ e_{22}= & {} 2\sum _{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-1}+i}a_{2^{n}-1-i}.\\ S^{2}= & {} |a_{0}a_{3}-a_{1}a_{2}|^{2}+|a_{0}a_{5}-a_{1}a_{4}|^{2} \\&+\,|a_{0}a_{7}-a_{1}a_{6}|^{2}+|a_{2}a_{5}-a_{3}a_{4}|^{2} \\&+\,|a_{2}a_{7}-a_{3}a_{6}|^{2}+|a_{4}a_{7}-a_{5}a_{6}|^{2}. \end{aligned}$$