Let P_1∈ U_1 otimes…otimes U_n and P_2∈ V_1 otimes…otimes V_n be persistent tensors, where U_1,…,U_n and V_1,…,V_n are vector spaces with dim U_k=d_k and dim V_k=d'_k, respectively. Kronecker-product conjecture. Their Kronecker product P_1 boxtimesP_2 is persistent, and hence operatornamerk(P_1 boxtimesP_2)≥∑_k=1^n(d_k+d'_k-1)+1. Persistent tensors provide lower bounds for tensor rank, with potential applications to tensor complexity and quantum information. The source says that many examples have been checked but leaves the proof as an open problem.

The Kronecker-product conjecture for persistent tensors

From papers

Let $\mathcal{P}_1\in U_1\otimes\cdots\otimes U_n$ and $\mathcal{P}_2\in V_1\otimes\cdots\otimes V_n$ be persistent tensors, where $U_1,\ldots,U_n$ and $V_1,\ldots,V_n$ are vector spaces with $\dim U_k=d_k$ and $\dim V_k=d'_k$ , respectively. Kronecker-product conjecture. Their Kronecker product $\mathcal{P}_1\boxtimes\mathcal{P}_2$ is persistent, and hence

\operatorname{rk}(\mathcal{P}_1\boxtimes\mathcal{P}_2)\geq\sum_{k=1}^{n}(d_k+d'_k-1)+1.

Persistent tensors provide lower bounds for tensor rank, with potential applications to tensor complexity and quantum information. The source says that many examples have been checked but leaves the proof as an open problem.

Progress summary

Solved

The conjecture is false in general: Shitov found a counterexample without symmetry, while a newer paper proves the corresponding symmetric statement.

Introduced in earlier work and presented as open in 2022, the conjecture asserted that the Kronecker product of any two persistent tensors remains persistent, with the stated lower bound on tensor rank.

2026 counterexample and symmetric theorem

A 2026 paper reports that Shitov constructed a nonsymmetric counterexample, so the unrestricted conjecture is false. The same paper proves that Kronecker products of symmetric persistent tensors remain persistent, including iterated products and powers; this is a theorem for the symmetric analogue, not for the original statement.

Current status (as of August 2026): The unrestricted conjecture is settled negatively by a reported nonsymmetric counterexample, while the symmetric analogue is proved.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Masoud Gharahi, “Classifying Entanglement by Algebraic Geometry”, arXiv:2408.12265 (2025).

Additional references

2 papers in this index state this conjecture (2022–2024). The statement above is taken from the most recent of them; the others are arXiv:2211.00652.

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