A strengthened 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 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.

Progress summary

Open

No public discussion or published progress was found for this conjecture.

No public discussion or published progress was found.

Current status (as of August 2026): The conjecture remains open, with no recorded public activity or verified progress.

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 $v\in V_1\otimes V_2\otimes V_3$ , 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.

0 endorsements

Shivam Patel · 1 day ago