Distillation of every Wigner-negative depolarized Strange state
- Field
- Topics
Problem
Can every Wigner-negative state in the depolarized qutrit Strange-state family be distilled arbitrarily accurately by catalyst-free stabilizer operations?
Define
For every \(0\leq\varepsilon<3/4\) and \(\delta>0\), can finitely many copies of the state in Eq. (1) be converted, with positive success probability, to a state within trace distance \(\delta\) of \(|S\rangle\langle S|\) using stabilizer ancillas, Clifford unitaries, Pauli measurements, classical feedforward, and discarding, but no magic catalyst?
Source
This precise formulation is editor wording based on the unresolved direction and limitations documented in the cited primary literature [Zurel26][Veitch12]; it is not presented as a verbatim conjecture of those authors.
Progress
For the family in Eq. (1), the Gross discrete Wigner function has one entry \((4\varepsilon-3)/9\) and all remaining entries \((3-\varepsilon)/18\). Thus the state is Wigner-negative exactly for \(\varepsilon<3/4\) [Veitch12][Prakash20].
Necessity is known. Wigner-positive states remain Wigner-positive under stabilizer protocols, including conditioning on allowed measurement outcomes. Odd-dimensional nonstabilizer states with nonnegative Wigner functions therefore give genuine bound magic. Their existence does not answer the question about the Wigner-negative region [Veitch12].
2020 construction. The 11-qutrit ternary Golay protocol distills the Strange state with cubic noise suppression. In the depolarizing convention used above, its threshold is approximately \(\epsilon=0.387\), well below \(0.75\) [Prakash20] (Sections 3 and 5).
February 3, 2026 revision; 2026 journal publication. Prakash and Singhal searched a large class of qutrit stabilizer codes up to length 23. They found over 600 length-23 CSS examples with cubic suppression, but none surpassed the Golay threshold. This was a substantial search, not an exhaustive theorem over every possible protocol [Prakash26].
March 2026 update. The quadratic-residue-code analysis continues to identify the Golay code as the best known Strange-state threshold and leaves the Wigner-negativity conjecture open [Zurel26].
Status: open, including this simple radial test case. The presently unclosed interval should not be labeled a proven Wigner-negative bound-magic region.
Comment
The allowed operation class is stabilizer operations, not all stabilizer-preserving channels, and no magic catalyst is free; these distinctions affect conversion theorems [Zurel26][Fang26].
Compared with the general mixed-state conjecture, this family offers a single real parameter, an exact geometric boundary, and a well-defined existing benchmark. It is a natural place to investigate whether coding constraints force a universal threshold gap or merely reflect limitations of existing constructions.
The appropriate negative result would rule out all adaptive stabilizer protocols, not just CSS projections or a bounded code-length search. A positive result along this ray would still leave other Wigner-negative qutrit directions to analyze.
References
- [Zurel26]
- M. Zurel, S. Jana, and N. de Silva, High-threshold magic state distillation with quantum quadratic residue codes, March 19, 2026. Introduction explicitly states the open conjectures; Sections 4.2–4.3 give the new constructions.link
- [Fang26]
- K. Fang and Z.-W. Liu, One-shot distillation with constant overhead using catalysts, Nature Communications 17, 6010 (2026); September 2, 2026. See Definitions 1–2 and Theorem 7.link
- [Veitch12]
- V. Veitch, C. Ferrie, D. Gross, and J. Emerson, Negative Quasi-Probability as a Resource for Quantum Computation, New Journal of Physics 14, 113011 (2012); Establishes the Wigner-positive obstruction and bound magic.arXiv
- [Prakash20]
- S. Prakash, Magic state distillation with the ternary Golay code, Proceedings of the Royal Society A 476, 20200187 (2020); arXiv:2003.02717. See the depolarizing-noise convention and Strange-state threshold.DOI
- [Prakash26]
- S. Prakash and R. Singhal, A Search for High-Threshold Qutrit Magic State Distillation Routines, February 3, 2026; published as Search for high-threshold qutrit magic-state distillation routines, Physical Review A 113, 042404 (2026). See the Introduction, code-search results, and conclusion.link