Detection of genuine n-qubit entanglement via the proportionality of two vectors

In [Science 340:1205, (2013)], via polytopes Michael Walter et al. proposed a sufficient condition detecting the genuinely entangled pure states. In this paper, we indicate that generally, the coefficient vector of a pure product state of n qubits cannot be decomposed into a tensor product of two vectors, and show that a pure state of n qubits is a product state if and only if there exists a permutation of qubits such that under the permutation, its coefficient vector arranged in ascending lexicographical order can be decomposed into a tensor product of two vectors. The contrapositive of this result reads that a pure state of n qubits is genuinely entangled if and only if its coefficient vector cannot be decomposed into a tensor product of two vectors under any permutation of qubits. Further, by dividing a coefficient vector into 2i equal-size block vectors, we show that the coefficient vector can be decomposed into a tensor product of two vectors if and only if any two nonzero block vectors of the coefficient vector are proportional. In terms of proportionality,we can rephrase that a pure state of n qubits is genuinely entangled if and only if there are two nonzero block vectors of the coefficient vector which are not proportional under any permutation of qubits. Thus, we avoid decomposing a coefficient vector into a tensor product of two vectors to detect the genuine entanglement. We also present the full decomposition theorem for product states of n qubits.