bibkey: lomonaco2025nonstabilizerness authors: Gabriele Lo Monaco; Salvatore Lorenzo; Alessandro Ferraro; Mauro Paternostro; G. Massimo Palma; Luca Innocenti year: 2025 title: “The non-stabilizerness cost of quantum state estimation” doi: 10.48550/arXiv.2510.00157 url: https://arxiv.org/abs/2510.00157v2 claim: “Conjecture 1: For any pure |ψ⟩, the maximal value of s_μ can be obtained with maximally entangled S.” strata_touched:

• D5/S3/Quantum/Measurement/StabilizerPovmMaximalEntanglementRefutation license: citation-only triage: anchor

The non-stabilizerness cost of quantum state estimation

G. Lo Monaco, S. Lorenzo, A. Ferraro, M. Paternostro, G. M. Palma and L. Innocenti, arXiv:2510.00157v2 (quant-ph), 27 October 2025.

Conjecture 1, Section IV, PDF page 8:

For any pure |ψ⟩, the maximal value of s_μ can be obtained with maximally entangled S.

Here n is the number of data qubits, m is the number of ancillas, and ψ is the fixed pure ancilla state. The maximum ranges over all stabilizer measurement bases on n + m qubits.

Section II, equation (1), PDF page 3:

μ_b = Tr_R[U† P_{Φ′b} U (𝕀 ⊗ ρ_R)] = (I ⊗ ⟨ψ|) P{Φ_b} (I ⊗ |ψ⟩)

P_{Φ_b} ≡ |Φ_b⟩⟨Φ_b|

Section II, PDF page 3:

s_μ ≡ dim span({μ_b}_{b=1}^{n_out})

The paper fixes n_out = 2^(n+m). The matrix span is over the complex numbers. The formal effective operator uses the frozen IsometricCompression.compress(ancillaEmbedding(ψ), A), with the ancilla frame I⊗|ψ⟩ constructed from Mathlib Matrix.kronecker and Matrix.replicateCol.

Section II, PDF page 4:

Equivalently, a stabilizer group can be defined as a maximal Abelian subgroup of Pₙ that does not contain −I.

We will instead talk of a stabilizer basis to refer to an orthonormal basis of pure (stabilizer) states that are the common eigenvectors of some stabilizer group.

Section IV, PDF page 7:

Sₙ, Sₘ ≤ S the subgroups of operators with support only on Hₙ and Hₘ, respectively

The parameter p quantifies the entanglement of the states and satisfies 2p = n + m − dim(Sₙ) − dim(Sₘ) and 0 ≤ p ≤ min(n, m).

States are separable for p = 0 and maximally entangled for p = min(n, m).

The dimensions of these finite elementary Abelian support subgroups are their binary dimensions, equivalently log₂ of their cardinalities.

Appendix A, PDF page 15:

The n-qubit Pauli group Pₙ is the group generated by the n-fold tensor products of the single-qubit Pauli matrices, {I, X, Y, Z}, along with multiplicative factors of ±1, ±i.

Refutation

The formal refutation uses n = m = 3 and the unit vector (1−i, 1+i, −2, −2, 2, 2, 2+2i, 2−2i)/6 in computational order 000 through 111. Every maximally entangled stabilizer basis has effective span dimension at most 31. A basis with two Bell pairs and local Z measurements on the remaining data and ancilla qubits has effective span dimension at least 32. These two inequalities suffice to refute Conjecture 1; no claim about other conjectures or an optimal value beyond this comparison is made.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2510.00157
• URL: https://arxiv.org/abs/2510.00157v2
• PDF: https://arxiv.org/pdf/2510.00157v2, read 2026-09-30: the definitions on pages 3–4, entanglement on page 7, Conjecture 1 on page 8, and the Pauli group on page 15. The preregistration in issue #11467 reports no settlement found in its literature search; that search does not prove absence of all later work.