Strange — Stabilizer Rank Challenge

The state

| S ⟩ = \frac{| 1 ⟩ - | 2 ⟩}{\sqrt{2}}

The Strange state. Its support misses |0⟩ entirely, which is what makes it unusual among the qutrit magic states: two of the nine amplitudes of |S⟩^⊗2 vanish, and the missing support is exactly what the rank-2 decomposition exploits.

Smallest known exact-rank exponent of the four qutrit orbits. Its m=3 and m=4 cells are settled at 4, so the exponent cannot be improved there; beating it needs m=6 at rank 7, where search terminates at exactly √(8/27) and never below.

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| S ⟩}^{\otimes 1}) \geq 2$	1	2	—	lean	@vprusso · stabrank verify_challenge
$χ_{R} ({\| S ⟩}^{\otimes 1}) \leq 2$	1	2	0.6309	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 2}) \geq 2$	2	2	—	lean	@vprusso · stabrank PR #19
$χ_{R} ({\| S ⟩}^{\otimes 2}) \leq 2$	2	2	0.3155	lean	@FarLab @vprusso · arXiv:2605.28586, Theorem 2
$χ_{R} ({\| S ⟩}^{\otimes 3}) \geq 4$	3	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 3}) \leq 4$	3	4	0.4206	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 4}) \geq 4$	4	4	—	reproduced	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 4}) \leq 4$	4	4	0.3155	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 5}) \geq 4$	5	4	—	reproduced	@vprusso · stabrank rank_exclusion.py
$χ_{R} ({\| S ⟩}^{\otimes 5}) \geq 5$	5	5	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| S ⟩}^{\otimes 5}) \geq 6$	5	6	—	attested · 3/4259 re-run	@vprusso · stabrank-research, round 33 (attacks/round33-strange-rank5)
$χ_{R} ({\| S ⟩}^{\otimes 5}) \leq 8$	5	8	0.3786	lean	@vprusso · product of board witnesses
$χ_{R} ({\| S ⟩}^{\otimes 6}) \geq 5$	6	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| S ⟩}^{\otimes 6}) \geq 6$	6	6	—	attested · 3/4259 re-run	@vprusso · stabrank-research, round 33 (attacks/round33-strange-rank5)
$χ_{R} ({\| S ⟩}^{\otimes 6}) \leq 8$	6	8	0.3155	lean	@FarLab @vprusso · arXiv:2605.28586
$χ_{R} ({\| S ⟩}^{\otimes 7}) \geq 5$	7	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| S ⟩}^{\otimes 7}) \geq 6$	7	6	—	attested · 3/4259 re-run	@vprusso · stabrank-research, round 33 (attacks/round33-strange-rank5)
$χ_{R} ({\| S ⟩}^{\otimes 7}) \leq 16$	7	16	0.3605	verified	@vprusso · stabrank research/constructions/product_witness.py
$χ_{R} ({\| S ⟩}^{\otimes 8}) \geq 5$	8	5	—	reproduced	@vprusso · stabrank verify_challenge
$χ_{R} ({\| S ⟩}^{\otimes 8}) \geq 6$	8	6	—	attested · 3/4259 re-run	@vprusso · stabrank-research, round 33 (attacks/round33-strange-rank5)
$χ_{R} ({\| S ⟩}^{\otimes 8}) \leq 16$	8	16	0.3155	verified	@vprusso · stabrank research/constructions/product_witness.py

What would move it

Beating the published exponent needs χ ≤ 7 at m=6→ 0.2952.