Let V_1,V_2,V_3 be the local qudit vector spaces with dimensions d_1,d_2,d_3, and let n_a be the corresponding one-party nullities. For a tripartite state v∈ V_1 otimes V_2 otimes V_3, let n_1,2,3 denote the nullity of the common kernel space K_1,2,3(v). The tripartite-nullity lower-bound conjecture. The nullity n_1,2,3 satisfies n_1,2,3≥max bigl(d_1d_2-d_3+n_3)n_3,(d_1d_3-d_2+n_2)n_2,(d_2d_3-d_1+n_1)n_1bigr. The authors report extensive empirical evidence for this lower bound, which is intended to improve the available estimates for the invariant controlling tripartite entanglement classes; its general validity remains open.

A lower-bound conjecture for the tripartite nullity

From papers

Let $V_1,V_2,V_3$ be the local qudit vector spaces with dimensions $d_1,d_2,d_3$ , and let $n_a$ be the corresponding one-party nullities. For a tripartite state $v\in V_1\otimes V_2\otimes V_3$ , let $n_{1,2,3}$ denote the nullity of the common kernel space $K_{1,2,3}(v)$ . The tripartite-nullity lower-bound conjecture. The nullity $n_{1,2,3}$ satisfies

n_{1,2,3}\ge\max\bigl\{(d_{1}d_{2}-d_{3}+n_{3})n_{3},(d_{1}d_{3}-d_{2}+n_{2})n_{2},(d_{2}d_{3}-d_{1}+n_{1})n_{1}\bigr\}.

The authors report extensive empirical evidence for this lower bound, which is intended to improve the available estimates for the invariant controlling tripartite entanglement classes; its general validity remains open.

Progress summary

Open

The conjectured lower bound has supporting experiments but no publicly reported proof or counterexample, so the problem remains open.

The problem asks whether a proposed lower bound always holds for the common-kernel nullity of a tripartite quantum state. It appears as Conjecture 3.2 in a December 2024 preprint, which presents it as an open question.

December 2024 preprint

The authors report extensive empirical evidence for the conjecture and describe the one-party nullity problem as completely characterized. For the tripartite common-kernel nullity, however, they give only bounds and no general proof; the retrieved sources report no counterexample, verification, or claimed solution.

Current status (as of August 2026): The lower bound remains an empirically supported open conjecture, with no publicly reported proof or counterexample in the retrieved sources.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Roman V. Buniy and Thomas W. Kephart, “Tripartite entanglement of qudits”, arXiv:2412.10728 (2024).

1 week ago

Let $v\in V_1\otimes V_2\otimes V_3$ , put $d_i=\dim V_i$ , $n_i=\dim K_i(v)$ , and set $r_i=d_i-n_i$ . Thus $r_i$ is the rank of the $i$ -th flattening of $v$ .

For each permutation $(i,j,k)$ of $(1,2,3)$ , the two-party contraction

f_{j,k}(v):V_j\otimes V_k\longrightarrow V_i^*

has rank $r_i$ , and hence

\dim\ker f_{j,k}(v)=d_jd_k-r_i.

For $u\in K_i(v)$ and $z\in\ker f_{j,k}(v)$ , the tensor $u\otimes z$ belongs to the common kernel $K_{1,2,3}(v)$ . Indeed, the $(j,k)$ -contraction vanishes because $z\in\ker f_{j,k}(v)$ , and each of the other two pair contractions vanishes because $u\in K_i(v)$ . Consequently,

K_i(v)\otimes\ker f_{j,k}(v)\subseteq K_{1,2,3}(v),

n_{1,2,3}(v)\geq n_i(d_jd_k-r_i) =n_i(d_jd_k-d_i+n_i).

Taking the maximum over $i=1,2,3$ gives precisely the asserted lower bound for every tripartite tensor, including the zero tensor.

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Shivam Patel · 1 day ago