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

About 2 years old · traced to 2024

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.

References

Primary source

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-posted argument claims a complete proof of the conjecture, but it has not been independently checked, so the result is not settled.

Buniy and Kephart proposed the conjecture in December 2024: the common-kernel nullity should exceed an explicit maximum determined by the dimensions and one-party nullities.

Known results

The authors report extensive empirical evidence, but state that they have only upper and lower bounds for $n_{1,2,3}$ and no complete result.

Posted attempt

A reader claims a complete proof by showing, for each permutation $(i,j,k)$ , that $K_i(v)\otimes\ker f_{j,k}(v)\subseteq K_{1,2,3}(v)$ ; dimension counting then gives each term in the conjectured maximum. The argument has not been independently verified.

Current status (as of August 2026): The conjecture has a complete-proof claim based on an unverified reader argument; no independently corroborated proof or counterexample is recorded.

Sources

arxiv.org

ProofThis solution needs a summarySee full solutionHide full solution

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.