bibkey: li2025genuinely authors: D. Li year: 2025 title: “A necessary and sufficient condition for genuinely entangled n-qubit states with six non-zero coefficients” doi: 10.48550/arXiv.2510.16561 url: https://arxiv.org/abs/2510.16561v1 claim: “Section 6 asks how many forms of the corresponding 2 by p coefficient matrix exist for a separable, not trivially separable n-qubit state with m = 2p non-zero coefficients (p prime), after recording two forms for m = 4 and four forms for m = 6.” strata_touched:

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Li, genuinely entangled n-qubit states with six non-zero coefficients

D. Li, A necessary and sufficient condition for genuinely entangled n-qubit states with six non-zero coefficients, arXiv:2510.16561v1 (18 October 2025; quant-ph).

Verified locator

DOI: 10.48550/arXiv.2510.16561. Primary version: https://arxiv.org/abs/2510.16561v1 (the only arXiv version). The TeX source iff-m=6--e-print.tex of v1 supplies the ordered basis states (Section 2, Eq. (1), label order), trivial separability (Section 2), the four $m=6$ forms (Section 2, labels mt-1, mt-2, mt-3, mt-5), the separable form for $m=2p$ (Section 6, label dis-1, printed as Eq. (120)) and the closing question (Section 6, lines 924–927).

Source statements

Ordered basis states (Section 2): “let $|\psi {⟩}_{1\cdots n}={\sum }_{i=1}^6{b}_i|B_i{⟩}_{1\cdots n}$, where $b_i\ne 0$, $i=1,..,6$ and $|B_i{⟩}_{1\cdots n}$ are basis states, where $B_i$ are binary numbers ${\epsilon }_1^{(i)}{\epsilon }_2^{(i)}\dots {\epsilon }_n^{(i)}$ […] and $B_1<B_2<B_3<B_4<B_5<B_6$.”

Trivial separability (Section 2): “When $|\psi {⟩}_{1\cdots n}=|0{⟩}_i|\phi {⟩}_{12\cdots n/i}$ ($|1{⟩}_i|\varphi {⟩}_{12\cdots n/i}$), it means the ith qubit is 0 (1) in each basis state […]. We call it a trivially separable state.”

The $m=6$ forms (Section 2): “Via the six coefficients $b_1,\cdots ,b_6$, we can make $6!(=720)$ $2$ by $3$ matrices with the entries $b_i$. We will show that the corresponding coefficient matrices $\Delta$ are of the following four forms”: $\left(\begin{array}{ccc}{b}_1& b_2& b_5\\ b_3& b_4& b_6\end{array}\right)$, $\left(\begin{array}{ccc}{b}_1& b_3& b_5\\ b_2& b_4& b_6\end{array}\right)$, $\left(\begin{array}{ccc}{b}_1& b_2& b_3\\ b_4& b_5& b_6\end{array}\right)$, $\left(\begin{array}{ccc}{b}_1& b_3& b_4\\ b_2& b_5& b_6\end{array}\right)$.

The case $m=2p$ (Section 6): “Let $|\psi {⟩}_{1\cdots n}$ be a pure state of $n$ qubits and $m$ be the number of non-zero coefficients of $|\psi {⟩}_{1\cdots n}$, where $m=2p$, where $p$ is a prime. […] Assume that $|\psi {⟩}_{1\cdots n}$ is not trivially separable. Then if $|\psi {⟩}_{1\cdots n}$ is separable, then it can be written as $(\alpha_1|0_1\gamma_2\cdots\gamma_k\rangle+\alpha_2|1_1\gamma_2’\cdots\gamma_k’\rangle){p_1\cdots p_k}\otimes (\beta_1|\sigma_1^{(1)}\cdots\sigma_s^{(1)}\rangle+\cdots+\beta_p|\sigma_1^{(p)}\cdots\sigma_s^{(p)}\rangle){q_1\cdots q_s}$”, with “the following corresponding coefficient matrix $\left(\begin{array}{ccc}{\alpha }_1{\beta }_1& \cdots & {\alpha }_1{\beta }_p\\ {\alpha }_2{\beta }_1& \cdots & {\alpha }_2{\beta }_p\end{array}\right)$.”

Question (Section 6): “What to do next is how to find the forms of the corresponding coefficient matrices. When $m=4$, there are two forms [Li-24]. When $m=6$, there are four forms. How many forms of the corresponding coefficient matrices are there when $m=2p$?”

Scope

The paper proves a necessary and sufficient condition for genuine entanglement of $n$-qubit states with six non-zero coefficients and lists the four $m=6$ forms. The count for general $m=2p$ is left as the closing question.