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slug: descamps-2024-binary-stabilizer-local-equivalence bibkey: descamps2024stabilizer doi: 10.1088/1751-8121/ad8607 url: https://arxiv.org/abs/2309.09815v1 triage: theorem motivation_gids:

  • D5/S3/Quantum/Information/BinaryStabilizerLocalInequivalence.result

A binary-stabilized six-qubit state that is not locally a stabilizer state

Problem

É. Descamps and B. Dakić (arXiv:2309.09815, quant-ph; J. Phys. A 57, 455301 (2024)) call a state of N qubits stabilized by a stabilizing set 𝒜 when it is, up to a complex factor, the unique common +1 eigenvector of finitely many tensor products of elements of 𝒜. Standard stabilizer states are those stabilized by 𝒫 = {±1, ±i}·{𝟙, X, Y, Z}. After an exhaustive analysis of two and three qubits the paper states:

\begin{conj} All states stabilized by the set $\mathcal A$ composed of binary
operators and the identity {\it i.e.}, $\mathcal{A}=\{A_{\theta,\phi},\1_2\}$,
are locally equivalent to standard stabilizer states (\textit{i.e.},
stabilized by the Pauli group
$\mathcal{P}=\{\pm 1,\pm i\}\cdot\{\1,\sigma_X,\sigma_Y,\sigma_Z\}$).
\end{conj}

Issue #11565 fixes the reading. A_{θ,φ} = cos θ Z + sin θ (cos φ X + sin φ Y), stabilization needs no commutativity, and local equivalence is ψ = (U_1 ⊗ ⋯ ⊗ U_N) φ with U_i ∈ U(2). The formal claim is the conjecture for every N; the result refutes it.

Motivation

If the conjecture held, binary stabilization would describe nothing beyond the standard stabilizer formalism up to local unitaries. The frozen declaration D5/S3/Quantum/Information/BinaryStabilizerLocalInequivalence.result shows that six qubits already give a state stabilized by four tensor products of binary operators that no local unitary maps to a standard stabilizer state.

Gap

Issue #11565 preregisters the reading, the witness, the route and the literature check. The conjecture is in the only arXiv version. Semantic Scholar lists 6 citing records; the arXiv sources of the five with identifiers were searched, and none treats the conjecture. Author and abstract searches on arXiv found no follow-up on binary stabilization. The paper notes that the analogous statement fails for the XS stabilizing set of Ni et al., which is not a set of binary operators.

These readings are not-found-in-searched-scope; they do not establish an exhaustive worldwide literature search, priority, or the absence of an independent answer.

Route

  1. Let v have coefficient 1 on the six labels of Hamming weight one, −1 on the six of weight five and 0 elsewhere, and take O_1 = (−X) ⊗ X^{⊗5}, O_2 = (−Z) ⊗ Z^{⊗5}, O_3 = H^{⊗6}, O_4 = K^{⊗6} with H = (X+Z)/√2 = A_{π/4,0} and K = (X+Y)/√2 = A_{π/2,π/4}.
  2. O_2 kills even weights, O_1 makes complementary coefficients opposite, O_4 kills weight three, and O_3 at six weight-three labels forces the weight-one coefficients to be equal; all four fix v.
  3. For a state φ stabilized by phased Pauli words, every Pauli word P anticommutes with some stabilizer T, giving ⟨φ, Pφ⟩ = −⟨φ, Pφ⟩ = 0, or commutes with all of them, giving Pφ = ±φ. So the pair purity Σ_{g,h} |⟨φ, (g ⊗ h ⊗ 𝟙^{⊗4}) φ⟩|² is an integer multiple of ‖φ‖⁴.
  4. The pair purity is 4 tr(ρ₁₂ ρ₁₂†) for the unnormalized reduced state of qubits 1, 2, and local unitaries conjugate ρ₁₂ by a unitary, so it and ‖φ‖ are invariant under local equivalence. For v the pair purity is 192 and ‖v‖⁴ = 144.

Falsifier

The answer would change if stabilization required the operators to commute, or if local equivalence allowed non-unitary local maps; the paper defines neither restriction. It would also change if binary operators excluded the signs −X, −Z; these are A_{π/2,π} and A_{π,0}.

Evidence

Exact SymPy arithmetic (issue #11565) gives O_a v = v, each O_a Hermitian with O_a² = 𝟙, a one-dimensional common +1 eigenspace, and the marginal ρ₁₂ of v/‖v‖ with tr(ρ₁₂²) = 1/3. As a positive control, the Pauli stabilizer state (|0⁶⟩ + |1⁶⟩)/√2 gives 4 tr(ρ₁₂²) = 2, an integer.

The canonical source is D5/S3/Quantum/Information/BinaryStabilizerLocalInequivalence.lean. Its public declarations are binaryOp, binarySet, pauliSet, StabilizedBy, LocallyEquivalent, claim and result. The frozen module state has statement identity sha256:6f7d09fe6589c9cd9f0d6519093013695a5d05a890462b95ea44ec07457b1c0e. The result declaration has statement identity sha256:1728416a009c8e60a1ff43fe1a0335498101aaec112556e56c78cb36403ce42d. The Freeze event is sha256:952fa2aa0a6dd3306b25ca5f89866102f8298d690aca9635216938b583b113b3. Its project-level frozen prerequisite is the module providing Pauli, pauliMatrix, tensorOp and wordOp, which itself builds on qubitX and qubitZ. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Triage

Tier 1 conjecture from a 2023 quant-ph paper, preregistered in issue #11565 before any Lean. theorem; resolution refuted. The public theorem has proof_shape: content, with escape witness form (2), the conclusion itself: inside its proof, the expectation trichotomy for Pauli stabilizer states and the local-unitary invariance of the pair purity are established as local steps on its live path. Its admission basis is open-problem-resolution. Utility kind=certified-instance; basis=refutes with typed claim and result.

What the settlement shows

Proved (Lean): the six-qubit state v, which is |W₆⟩ − X^{⊗6}|W₆⟩ for the W state |W₆⟩ up to normalization, is stabilized by four tensor products of binary operators and is not locally equivalent to any state stabilized by phased Pauli words.

Mechanism (proved inside result for qubits 1 and 2): for a state φ stabilized by phased Pauli words, every Pauli word P has ⟨φ, Pφ⟩ ∈ {0, ±‖φ‖²}, so the unnormalized pair purity Σ_{g,h} |⟨φ, (g ⊗ h ⊗ 𝟙^{⊗4}) φ⟩|² is an integer multiple of ‖φ‖⁴, and local unitaries preserve it and ‖φ‖. For v it is 192 with ‖v‖⁴ = 144; in normalized form, 4 tr(ρ₁₂²) = 4/3 for ρ₁₂ the two-qubit marginal of v/‖v‖, while every Pauli stabilizer state gives an integer.

Computed (NumPy, not formalized): every two-qubit marginal of v has spectrum (1/3, 1/3, 1/3, 0), every one-qubit marginal is maximally mixed and every three-qubit marginal has purity 1/4. The six-qubit XS-stabilizer state of Ni et al. that the paper cites has maximally mixed two-qubit marginals (purity 1/4), so v is not locally equivalent to it either.

Open: the weaker conjecture that the paper states for binary operators in the (XZ)-plane (A = {A_θ, 1}); the witness uses K = (X + Y)/√2, which is not in that plane. Also open: whether a counterexample exists on four or five qubits (the paper’s exhaustive analysis covers two and three qubits).

Source consequences: the paper’s theorem on generalized Clifford groups and its two- and three-qubit analysis do not depend on the conjecture and stand; the stabilization by binary operators and the identity is strictly larger, up to local equivalence, than the Pauli stabilizer formalism.

ASSUMED-UNVERIFIED

The published J. Phys. A text was not read, so it is unverified whether the conjecture was edited there. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent answer. The Lean kernel does not authenticate the external source or its version history.