χ_R(∣T3⟩^⊗2) ≤ 3

T₃ orbit, qutrit.

Participate

Run it yourself, or point a coding agent at it.

git clone https://github.com/unitaryfoundation/stabrank
cd stabrank
# describe your decomposition in bounds/mybound.json
make fit BOUND=bounds/mybound.json ARGS=--write
make verify BOUND=bounds/mybound.json

fit solves for the exact coefficients, or tells you no exact combination of your terms reproduces the target. verify rebuilds the identity in exact arithmetic. If it passes, open a pull request adding the file and the leaderboard picks it up on the next build.

A bound the pipeline cannot check is recorded as cited and never takes a record.

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Verification

lean t3_m2_stabRank_le_three in LeanProofs.T3M2StabRank, machine-checked by Lean

Implied per-copy exponent γ ≤ 0.5000 against a published 0.5000, no improvement.

Lean proof

This bound is backed by a machine-checked theorem: t3_m2_stabRank_le_three in LeanProofs.T3M2StabRank. The recorded lake build confirms it compiles.

Notes

Tight: the matching lower bound is certified in this repository. The carry decomposition is formalized in Lean as t3_m2_vector_decomposition, which proves the pointwise identity at all nine points of F₃^2 and depends on no axioms beyond propext, Classical.choice and Quot.sound. The decomposition is due to Kocia and Sarovar; the formalization is this repository's. Now stated against stabRank: t3_m2_stabRank_le_three proves Stabilizer.stabRank IsStab |T3⟩^⊗2 ≤ 3, with each carry block shown to be a stabilizer state in the concrete sense (the line {(t, s - t)} with phase t^2, 2t^2 + t, or 0 for s = 0, 1, 2) rather than asserted in a comment; with t3_m2_stabRank_gt_two this gives t3_m2_stabRank_eq_three, χ(|T3⟩^⊗2) = 3 as one theorem about stabRank.

Attribution

Kocia and Sarovar · arXiv:2003.01130 · literature

Submission

JSON

{
  "schema_version": "0.1",
  "orbit": "T3",
  "m": 2,
  "direction": "upper",
  "rank": 3,
  "provenance": {
    "author": "Kocia and Sarovar",
    "reference": "arXiv:2003.01130",
    "method": "literature",
    "date": "2020-03-02",
    "github": []
  },
  "notes": "Tight: the matching lower bound is certified in this repository. The carry decomposition is formalized in Lean as t3_m2_vector_decomposition, which proves the pointwise identity at all nine points of F_3^2 and depends on no axioms beyond propext, Classical.choice and Quot.sound. The decomposition is due to Kocia and Sarovar; the formalization is this repository's. Now stated against stabRank: t3_m2_stabRank_le_three proves Stabilizer.stabRank IsStab |T3>^2 <= 3, with each carry block shown to be a stabilizer state in the concrete sense (the line {(t, s - t)} with phase t^2, 2t^2 + t, or 0 for s = 0, 1, 2) rather than asserted in a comment; with t3_m2_stabRank_gt_two this gives t3_m2_stabRank_eq_three, chi(|T3>^2) = 3 as one theorem about stabRank.",
  "lean": {
    "module": "LeanProofs.T3M2StabRank",
    "theorem": "t3_m2_stabRank_le_three"
  }
}