Hilbert–Schmidt inradius of the multiqubit stabilizer polytope
- Field
- Topic
Problem
Is every \(n\)-qubit state of purity at most \(1/(2^n-1/2)\) a convex mixture of stabilizer states?
Let \(d=2^n\) and
Is the following implication valid for every \(n\geq1\)?
Equivalently, does the Hilbert–Schmidt ball about \(I/d\) contained in Eq. (1) have radius \(1/\sqrt{d(2d-1)}\)? This is the threshold asserted by Eq. (2).
Source
Liu and coauthors pose the purity threshold as Conjecture 1 [Liu26]. Zurel and Davis give equivalent dual and inradius formulations in Conjecture 2 and Theorem 7 [Zurel26].
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Boundary construction. The Triangle Criterion supplies magic arbitrarily close to purity \(1/(d-1/2)\), but this is one side of the sharp-threshold question, not a proof that every lower-purity state is stabilizer [Liu26] (Theorem 5 and Conjecture 1).
February 25, 2026. Zurel and Davis obtain the same qubit inradius conditionally on their dual-purity conjecture and verify the relevant conjecture for one and two qubits. Their unconditional odd-prime-dimensional inradius result must not be mistaken for an unconditional multiqubit result [Zurel26].
September 3, revised September 8, 2026. Liu and Liu prove the stronger universal guarantee
\begin{equation} \operatorname{Tr}(\rho^2)\leq\frac1{d-a_*} \quad\Longrightarrow\quad\rho\in\mathrm{STAB}_n, \qquad a_*\simeq0.458327. \tag{3} \end{equation}The displayed definitions, constraints, and target bounds are recorded in Eqs. (3).
Thus the remaining dimension-independent gap is between \(a_*\simeq0.458327\) and \(1/2\). Their revised discussion explicitly leaves the exact inradius open [LiuLiu26] (Theorem 1 and Discussion).
Status: open, with very recent primary-source confirmation. Reporting only the older \(1/(d-1/d)\) sufficient bound would miss substantial 2026 progress.
- Reported progress: Issue #102 (withdrawn).
Comment
This formulation asks where magic first becomes possible near the maximally mixed state, without selecting a decoder, a target state, or an asymptotic computational model.
The dual formulation suggests a concrete mathematical attack: constrain the squared Hilbert–Schmidt norm of every trace-one operator nonnegative on stabilizer vectors. Searching for a feasible \(A\) with \(\operatorname{Tr}(A^2)>2\) would directly falsify the conjecture; numerical absence of such operators would not prove it. The negative eigenvalues allowed in \(\mathcal L_n\) are the source of difficulty, not a defect in the formulation.
References
- [Liu26]
- Z. Liu, T. Haug, Q. Ye, Z.-W. Liu, and I. Roth, Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection, April 20, 2026. See Theorem 5, Conjecture 1, and the discussion of arbitrary-copy bound magic.link
- [Zurel26]
- M. Zurel and J. Davis, Basis-independent stabilizerness and maximally noisy magic states, February 25, 2026. See Conjecture 2, Theorem 7, and Corollary 3.link
- [LiuLiu26]
- Z. Liu and Z.-W. Liu, On the geometry and typicality of quantum magic, September 8, 2026; first submitted September 3. See Theorem 1 and Discussion.link