A strengthened 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 strengthened tripartite-nullity lower-bound conjecture. The nullity $n_{1,2,3}$ satisfies

\begin{aligned} n_{1,2,3}&\ge n_{1}\bigl((d_{2}-n_{2})(d_{3}-n_{3})-(d_{1}-n_{1})\bigr)+n_{2}\bigl((d_{1}-n_{1})(d_{3}-n_{3})-(d_{2}-n_{2})\bigr)\\ &\quad+n_{3}\bigl((d_{1}-n_{1})(d_{2}-n_{2})-(d_{3}-n_{3})\bigr)\\ &\quad+\max\bigl\{n_{1}(d_{2}n_{3}+n_{2}d_{3}-n_{2}n_{3}),n_{2}(d_{1}n_{3}+n_{1}d_{3}-n_{1}n_{3}),n_{3}(d_{1}n_{2}+n_{1}d_{2}-n_{1}n_{2})\bigr\}\\ &\quad+\min\bigl\{(d_{1}-n_{1})n_{2}n_{3},n_{1}(d_{2}-n_{2})n_{3},n_{1}n_{2}(d_{3}-n_{3})\bigr\}. \end{aligned}

The additional minimum term strengthens the preceding empirical lower bound and is proposed as part of the authors' conjectural description of the tripartite nullity. 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 claims a complete proof of the strengthened conjecture, but no independent verification has been found.

The conjecture proposes an additional minimum term in the lower bound for the common kernel of a tripartite tensor. Buniy and Kephart introduced this conjectural bound in their 2024 study of tripartite qudit entanglement.

Posted attempt

A reader claims a complete proof by decomposing the tensor into active and null-coordinate sectors, obtaining an exact formula whose remaining term is a nonnegative common-kernel dimension. The argument has not been independently verified, so it establishes only a claimed resolution.

Current status (as of August 2026): A complete proof has been claimed in discussion, but the conjecture remains mathematically unsettled pending independent verification.

Sources

arxiv.org

ProofThis solution needs a summarySee full solutionHide full solution

Let $v\in V_1\otimes V_2\otimes V_3$ , let $d_i=\dim V_i$ , let $n_i=\dim K_i(v)$ , and set $r_i=d_i-n_i$ . Thus $r_i$ is the rank of the $i$ -th flattening of $v$ . Choose a basis in each factor and its corresponding dual basis so that $v$ is supported on

R_1\otimes R_2\otimes R_3,\qquad \dim R_i=r_i,

with complementary null-coordinate spaces $N_i$ of dimensions $n_i$ . Let $\kappa(v)\geq0$ be the common-kernel dimension of the three pair-contraction maps restricted to the active spaces.

Splitting the tensor in the common-kernel equations according to the eight coordinate sectors of

(R_1\oplus N_1)\otimes(R_2\oplus N_2)\otimes(R_3\oplus N_3)

gives the exact identity

\begin{aligned} n_{1,2,3}(v) ={}&\kappa(v) +n_1(r_2r_3-r_1)+n_2(r_1r_3-r_2)+n_3(r_1r_2-r_3)\\ &+n_1n_2r_3+n_1r_2n_3+r_1n_2n_3+n_1n_2n_3. \tag{1} \end{aligned}

Indeed, the all-active sector contributes $\kappa(v)$ . In the sector $N_1\otimes R_2\otimes R_3$ , the contractions involving the first factor vanish automatically, while the remaining contraction has rank $r_1$ on $R_2\otimes R_3$ , independently for each of the $n_1$ null coordinates. This sector therefore contributes $n_1(r_2r_3-r_1)$ ; the other one-null sectors follow by symmetry. Every sector containing at least two null factors is annihilated by all three pair contractions, giving the four remaining terms. Different support sectors have disjoint active/null output blocks, so these contributions form a direct sum.

Set

L=\sum_{\{i,j,k\}=\{1,2,3\}}n_i(r_jr_k-r_i),\quad a=r_1n_2n_3,\quad b=n_1r_2n_3,\quad c=n_1n_2r_3,\quad \tau=n_1n_2n_3.

Then (1) becomes

n_{1,2,3}(v)=\kappa(v)+L+a+b+c+\tau. \tag{2}

The three expressions inside the maximum in the conjectured bound are respectively

b+c+\tau,\qquad a+c+\tau,\qquad a+b+\tau,

and those inside its minimum are $a,b,c$ . Thus

\max\{b+c+\tau,a+c+\tau,a+b+\tau\}+\min\{a,b,c\} =a+b+c+\tau.

Consequently the entire conjectured lower bound is exactly $L+a+b+c+\tau$ . Formula (2) shows that the actual tripartite nullity exceeds this bound by precisely $\kappa(v)\geq0$ . Therefore the strengthened conjecture holds for every tripartite tensor, and (1) gives the exact additional nonnegative term.