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An improved lower-bound conjecture for the tripartite nullity

Let V1,V2,V3V_1,V_2,V_3 be the local qudit vector spaces with dimensions d1,d2,d3d_1,d_2,d_3, and let nan_a be the corresponding one-party nullities. For a tripartite state v∈V1⊗V2⊗V3v\in V_1\otimes V_2\otimes V_3, let n1,2,3n_{1,2,3} denote the nullity of the common kernel space K1,2,3(v)K_{1,2,3}(v). The improved tripartite-nullity lower-bound conjecture. The nullity n1,2,3n_{1,2,3} satisfies

n1,2,3≥n1((d2−n2)(d3−n3)−(d1−n1))+n2((d1−n1)(d3−n3)−(d2−n2))+n3((d1−n1)(d2−n2)−(d3−n3))+max⁡{n1(d2n3+n2d3−n2n3),n2(d1n3+n1d3−n1n3),n3(d1n2+n1d2−n1n2)}.\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\}. \end{aligned}

This is presented as a stronger empirically supported lower bound for the tripartite nullity and hence for the analysis of tripartite entanglement classes. Its validity remains open.

Progress summary

Solved

A 2024 paper left the stronger bound open, while a posted argument claims a complete proof that has not been independently verified.

Buniy and Kephart proposed the improved lower bound in 2024 for the common nullity of a tripartite tensor, strengthening earlier bounds. The paper records it as Conjecture 3.4 and leaves its validity unresolved.

Known results

  • Buniy and Kephart (2024) report extensive empirical evidence for the lower-bound conjectures.
  • The allowed values of the three one-party nullities are characterized completely, but only lower and upper bounds are known for n1,2,3n_{1,2,3}.

Posted attempt

A reader-written argument claims a complete proof: after decomposing into active and null sectors, it derives an identity reducing the conjectured inequality to κ(v)+min⁡{a,b,c}≥0\kappa(v)+\min\{a,b,c\}\geq 0. This attempt has not been independently verified.

Current status (as of August 2026): The conjecture has empirical support and a posted unverified proof attempt, but no independently verified proof or counterexample is recorded.

Sources
Sources & referencesView supporting material

Primary source

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

linear-algebraquantum-entanglementqudits

Solutions 1

Proof

Let v∈V1⊗V2⊗V3v\in V_1\otimes V_2\otimes V_3, let di=dim⁡Vid_i=\dim V_i, let ni=dim⁡Ki(v)n_i=\dim K_i(v), and set ri=di−nir_i=d_i-n_i. Thus rir_i is the rank of the ii-th flattening of vv. Choose a basis in each factor and its corresponding dual basis so that vv is supported on

R1⊗R2⊗R3,dim⁡Ri=ri,R_1\otimes R_2\otimes R_3,\qquad \dim R_i=r_i,

with complementary null-coordinate spaces NiN_i of dimensions nin_i. Let κ(v)≥0\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

(R1⊕N1)⊗(R2⊕N2)⊗(R3⊕N3)(R_1\oplus N_1)\otimes(R_2\oplus N_2)\otimes(R_3\oplus N_3)

gives the exact identity

n1,2,3(v)=κ(v)+n1(r2r3−r1)+n2(r1r3−r2)+n3(r1r2−r3)+n1n2r3+n1r2n3+r1n2n3+n1n2n3.(1)\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 κ(v)\kappa(v). In the sector N1⊗R2⊗R3N_1\otimes R_2\otimes R_3, the contractions involving the first factor vanish automatically, while the remaining contraction has rank r1r_1 on R2⊗R3R_2\otimes R_3, independently for each of the n1n_1 null coordinates. This sector therefore contributes n1(r2r3−r1)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=∑{i,j,k}={1,2,3}ni(rjrk−ri),a=r1n2n3,b=n1r2n3,c=n1n2r3,τ=n1n2n3.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

n1,2,3(v)=κ(v)+L+a+b+c+τ.(2)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+τ,a+c+τ,a+b+τ.b+c+\tau,\qquad a+c+\tau,\qquad a+b+\tau.

Therefore their maximum equals

a+b+c+τ−min⁡{a,b,c}.a+b+c+\tau-\min\{a,b,c\}.

Subtracting the proposed lower bound from (2) leaves exactly

κ(v)+min⁡{a,b,c}≥0.\kappa(v)+\min\{a,b,c\}\geq0.

Hence the conjectured inequality holds for every tripartite tensor.

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