Does a local QFT canonically supply the free structure that magic requires — or is magic irreducibly frame-dependent?

Statement

Magic is not a property of a state on an algebra. It is a property of an enriched object $(\mathcal{M},\mathcal{W},\mathfrak{F},\mathfrak{O},\omega )$ — algebra, distinguished Weyl (Pauli) system, free-state set, free-operation class, state. In finite dimensions the enrichment is handed to you: $\mathcal{W}=$ the Pauli group, $\mathfrak{F}=$ the stabilizer polytope, $\mathfrak{O}=$ stabilizer protocols. In a local QFT nothing hands it to you.

The question: does a Haag–Kastler net (haag-kastler-axioms) possess a canonical enrichment — one compatible with isotony, covariance, local normality, and the scaling limit — that can play the role of the Pauli/Clifford datum? Or can one prove that any such choice is necessarily regulator-, representation-, or model-dependent, and characterize the minimal extra datum that must be declared?

Sharply: modular theory supplies the divergences. It does not supply $\mathcal{W}$.

Why It Matters

Benedetti–Dabholkar–Dalmonte established that every physical state of a local QFT is necessarily magical — a witness, forced by the continuous Bisognano–Wichmann spectrum and type III₁ structure. The natural next move is to make it quantitative: how much magic? That move is blocked, and this question is the blockage.

• The block is not technical difficulty. It is a no-go: no automorphism-invariant functional of $(B(\mathcal{H}),\rho )$ can be a faithful magic measure, because every automorphism of a finite type-I factor is inner while stabilizerness is invariant only under the Clifford normalizer. One qubit already settles it (see below).
• Type III₁ makes the situation worse for quantification, not better: if every faithful normal state has the same full positive modular spectral support, that support cannot order states by magic.
• Answering it either way is publishable. A canonical enrichment would be a genuinely new structure in AQFT; a proof that none exists, with the minimal declared datum characterized, is a clean structural theorem of exactly the kind the group’s type III₁ programme produces.
• It is the same problem as OQP 2.5 (free-states-and-operations-for-resource-theory-in-qft), reached independently from the magic side rather than the resource-theory side. That convergence is itself evidence the problem is the right one.

What We Know

The no-go is exact and elementary. Let ${\rho }_{ϵ}^{+}=(1-ϵ)|+⟩⟨+|+ϵ1/2$ and ${\rho }_{ϵ}^A=T{\rho }_{ϵ}^{+}{T}^{†}$ with $T=\mathrm{diag}(1,e^{i\pi /4})$. Bloch vectors $(1-ϵ)\hat{x}$ and $\frac{1-ϵ}{\sqrt{2}}(1,1,0)$: the first sits inside the stabilizer octahedron always, the second outside for $0<ϵ<1-1/\sqrt{2}$. Identical spectra, unitarily equivalent modular operators, opposite verdicts. Qualification: in a fixed representation with a fixed basis, ${\Delta }_{\rho }$ can encode $\rho$ and hence its magic — but the basis has then silently supplied $\mathcal{W}$. The obstruction is to an intrinsic, covariant definition, which is precisely what AQFT should demand.

The obvious candidate for $\mathfrak{F}$ fails. Replacing “stabilizer” by “quasifree” — the natural Pauli $\to$ Weyl, stabilizer $\to$ Gaussian dictionary — defines $N_O(\omega )={\inf }_{\phi \in {\mathfrak{F}}_O^{\mathrm{qf}}}{S}_{\mathcal{A}(O)}(\omega \Vert \phi ),$ a perfectly good algebraic quantity. But a free-field vacuum is quasifree, so $N_O({\omega }_0)=0$ — while that same vacuum is non-stabilizer by the anti-flatness criterion. So $N_O$ is relative modular non-Gaussianity, a third resource, not magic. Any proposal must survive this test.

What survives the passage to type III. The divergence machinery does: Araki relative entropy and Petz–Rényi divergences are built from ${\Delta }_{\phi |\omega }$, so $M_{\mathfrak{F}}(\omega )={\inf }_{\phi \in \mathfrak{F}}[-⟨{\xi }_{\omega },\log {\Delta }_{\phi |\omega }{\xi }_{\omega }⟩]$ is well-posed the moment $\mathfrak{F}$ is named, and positivity, faithfulness, convexity, and monotonicity under free channels all follow from general resource theory. That is the template. The content is entirely in $\mathfrak{F}$.

What does not survive. Mana needs a discrete Wigner frame and does not extend. Generalized robustness extends only order-theoretically, as ${\inf }_{\phi }{D}_{\max }$, not as a spectral expectation. The stabilizer Rényi entropy rewrite $M_{\alpha }=\log d-D_{\alpha }(q_{\psi }\Vert u)$ on ${\ell }^{\infty }({\mathcal{P}}_d)$ is exact but breaks at the first step in the continuum: an infinite Weyl group has no uniform counting measure.

Full development, benchmarks, and the claim ledger: quantum-magic-modular-qft-study.

Possible Approaches

1. Net-compatible axioms for ${\mathfrak{F}}_O$ (highest value). Propose free-state families satisfying isotony under restriction, covariance, local normality, lower semicontinuity, stability under a declared class of local channels, and compatibility with spacelike composition. Then either exhibit one or prove the axioms are jointly unsatisfiable. Either outcome is a theorem.
2. Split inclusions as the regulator. For $\mathcal{A}(O_1)\subset \mathcal{N}\subset \mathcal{A}(O_2)$ with $\mathcal{N}$ type I, the split property supplies a genuine tensor factorization — hence a candidate finite Weyl frame. The question is whether the collar limit is choice-independent. Main risk, and it is the whole game: dependence on which split factor. This is the same route the OQP 2.5 page proposes; the two should be worked together.
3. Formalize the three-way non-equivalence. Turn “stabilizer magic $\ne$ non-Gaussianity $\ne$ modular anti-flatness” into a proposition with the free-field vacuum as the explicit separating example. Mathematically elementary, conceptually load-bearing, and it protects every downstream claim from the naming slippage the literature is already showing.
4. Weyl-characteristic projective limit. Replace the commutative Pauli-distribution picture by finite Weyl subspaces and take a projective limit. High novelty, low feasibility — the missing uniform measure is a real obstruction, not a technicality.
5. Quantify the witness instead. Ask whether regulator-independent Rényi mutual-information differences (the anti-flatness witnesses) bound any operational magic cost. This keeps the frame-independence that makes the Benedetti result attractive, at the price of giving up faithfulness.

Related Questions

• free-states-and-operations-for-resource-theory-in-qft — OQP 2.5; the same question from the resource-theory side. Work jointly.
• relative-entropy-magic-inequality — downstream: any magic ↔ relative-entropy ↔ Bell inequality must first name its free set.
• magic-of-coherent-squeezed-cat-states — the concrete computation that this question tells you how to interpret (and how not to).
• resource-theory-monotones-in-qft, holographic-resource-theory — sibling OQP resource-theory questions.
• quantum-information-in-infinite-dimensions — the umbrella problem (OQP 5.1).
• magic-and-confinement-phases, magic-and-the-gribov-horizon — applications that need a quantitative magic, hence need this answered first.