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Lower bounds for the quantum permanent

A mix of human-written and AI-generated textHuman understanding: some partsmath.OC — Optimization and Controlmath.FA — Functional Analysisquant-ph — Quantum Physicsmath.CO — Combinatorics

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

Griffiths's 1969 conjecture on the positivity of Schur forms reduces to the positivity of the quantum permanent Φn\Phi_n on positive bistochastic maps. This conjecture was refuted by Du with a map for which Φ9<0\Phi_9<0. Therefore, we ask: how negative can Φn\Phi_n be? We prove a lower bound Φn≥−3n/(3n2+20n+103)\Phi_n\ge-3n/(3n^2+20n+103) (for n≥8n\ge 8), and an asymptotically stronger bound Φn≥−1/n+(3e2−7−o(1))log⁡n/n2\Phi_n\ge-1/n+(3e^2-7-o(1))\log n/n^2. The key to our bounds is the orbital polynomial of the map, which turns Φn\Phi_n into a linear function of a distribution on partitions and positivity into linear constraints. We also construct explicit maps for n=8,9n=8, 9 that improve on Du's example. Finally, if d∗∈{4,…,8}d_*\in\{4,\ldots,8\} is the first dimension for which Φn<0\Phi_n<0, the minimum μn\mu_n of Φn\Phi_n over positive bistochastic maps satisfies −μn=Ω(n−d∗)-\mu_n=\Omega(n^{-d_*}); we conjecture the matching order −μn=O(n−d∗)-\mu_n=O(n^{-d_*}).

Provenance statement

The results in this paper were developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The author also used AI assistants (GPT and Claude) to help polish the writing. The proofs in Appendices B and C (the coefficient relaxations and the negative constructions) have not yet been checked line by line by the author.

Tools used

OpenAI
GPT Pro
Anthropic
Claude

References

  1. A. D. Alexandrov. Zur T heorie der gemischten V olumina von konvexen K örpern. IV : D ie gemischten D iskriminanten und die gemischten V olumina. Rec. Math. [Mat. Sbornik] N.S. , 3(45): 0 227--251, 1938.
  2. Heinz-Peter Breuer. Optimal entanglement criterion for mixed quantum states. Physical Review Letters , 97: 0 080501, 2006. 10.1103/PhysRevLett.97.080501. https://arxiv.org/abs/quant-ph/0605036v2.DOI
  3. Yun-Heng Du. G riffiths positivity does not imply positivity of the top C hern form, 2026. arXiv:2610.05931v1. https://arxiv.org/abs/2610.05931v1.arXiv
  4. Siarhei Finski. On characteristic forms of positive vector bundles, mixed discriminants, and pushforward identities. Journal of the London Mathematical Society , 106 0 (2): 0 1539--1579, 2022. 10.1112/jlms.12605.DOI
  5. Phillip A. Griffiths. Hermitian differential geometry, Chern classes, and positive vector bundles. In Donald C. Spencer and Shokichi Iyanaga, editors, Global Analysis (Papers in Honor of K. Kodaira) , pages 185--251. University of Tokyo Press, Tokyo, 1969. Author copy: https://publications.ias.edu/sites/default/files/hermdiffgeometrychern.pdf.link
  6. Leonid Gurvits. Quantum matching theory, 2002. arXiv:quant-ph/0201022v2. https://arxiv.org/abs/quant-ph/0201022v2.arXiv
  7. Leonid Gurvits. Classical complexity and quantum entanglement. Journal of Computer and System Sciences , 69 0 (3): 0 448--484, 2004. 10.1016/j.jcss.2004.06.003. Numbered locators refer to the preprint Classical deterministic complexity of Edmonds' problem and quantum entanglement , https://arxiv.org/abs/quant-ph/0303055v1.DOI
  8. Leonid Gurvits. The van der W aerden conjecture for mixed discriminants. Advances in Mathematics , 200 0 (2): 0 435--454, 2006. 10.1016/j.aim.2004.12.002. Numbered locators refer to the preprint, https://arxiv.org/abs/math/0406420v1.DOI
  9. William Hall. Constructions of indecomposable positive maps based on a new criterion for indecomposability, 2006. arXiv:quant-ph/0607035v1. https://arxiv.org/abs/quant-ph/0607035v1.arXiv
  10. Ian G. Macdonald. Symmetric Functions and Hall Polynomials . Oxford University Press, Oxford, 2nd edition, 1995.
Show all 12 referencesShow fewer references
  1. Suvrit Sra. Weak positivity of S chur forms under decomposable curvature, 2026. Preprint, https://github.com/suvrit/preprints.link
  2. Xueyuan Wan. Positivity of the third C hern form for G riffiths positive vector bundles, 2026. arXiv:2601.10424v2. https://arxiv.org/abs/2601.10424v2.arXiv

Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026