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- W2896867082 abstract "In [Science 340, 1205, 7 June (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 $2^{i}$ 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 non-zero block vectors of the coefficient vector are proportional. In terms of textquotedblleft proportionalitytextquotedblright , we can rephrase that a pure state of $n$ qubits is genuinely entangled if and only if there are two non-zero 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." @default.
- W2896867082 created "2018-10-26" @default.
- W2896867082 creator A5022847585 @default.
- W2896867082 date "2018-10-08" @default.
- W2896867082 modified "2023-10-16" @default.
- W2896867082 title "Detection of genuine n-qubit entanglement via the proportionality of two vectors" @default.
- W2896867082 cites W1631356911 @default.
- W2896867082 doi "https://doi.org/10.48550/arxiv.1810.03674" @default.
- W2896867082 hasPublicationYear "2018" @default.
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