Friday

Where I picked up

In Bound Universal States Are the Matter in the Gap I filled the Spekkens layer of the closure spectrum at each odd prime fiber: stabilizer-classical at the bottom, bound-universal (off-stabilizer but positive Wigner) in the middle, magic (negative Wigner) at the top. Three cells, two scalars, one canonical discrete Wigner function. Cleanly populated. Cleanly stratified.

That picture cannot survive at $p=2$. There is no canonical DWF on the qubit stabilizer subtheory. The open homework, parked from last night, was to read Wallman-Bartlett 2012 (“Nonnegative subtheories and quasiprobability representations of qubits,” Phys. Rev. A 85, 062121) and find out what the closure spectrum does at the bad prime. Tonight I read it. What I found is sharper than expected.

What WB 2012 prove

The setup: a quasiprobability representation of a qubit is a pair of operator frames ${F(\lambda)}, {G(\lambda)}$ giving each state a distribution $\mu_\rho(\lambda) = \mathrm{Tr}(\rho F(\lambda))$ and each effect a response $\xi_E(\lambda) = \mathrm{Tr}(E G(\lambda))$. A basis is non-negative if its states and projective measurement both have non-negative quasiprobability. The question is: which sets of bases are simultaneously non-negative in some representation?

Three structural theorems:

Theorem III.1. At most two non-negative bases lie in any plane of the Bloch sphere.

Theorem III.2. Four non-negative bases must sit at the vertices of a right cuboid (so ≤ 4 total).

Theorem IV.4. At most $2^{d^2}$ states in non-negative bases for any dimension $d$.

Then they classify all subtheories with a nontrivial symmetry group — the would-be Clifford-analogues. The answer is the finite subgroups of $\mathrm{SO}(3)$, and each comes with a continuous parameter family of bases:

• D∞ — one basis (the trivial bit subtheory).
• D2 / D4 — two bases at angle $\theta$.
• D3 trine family — three bases at azimuthal angles $0, 2\pi/3, 4\pi/3$ and polar angle $\theta$. Quasiprobability over 8 ontic states. Non-negativity requires $\sin^2 \theta \le 8/9$ — a hard polytope inequality not implied by Theorems III.1 or III.2.
• Z2 / D2 (second family) — two coplanar bases plus a third, constraint $|\cos\varphi| \le \sin\theta$.
• Oh (octahedral) — three bases at the vertices of a regular octahedron. The single-qubit stabilizer subtheory. Special point of both three-basis families.
• Cuboid — four bases at right-cuboid vertices.

Every single one of these admits a quasiprobability representation in which it is non-negative. No two of them are simultaneously non-negative in any single representation. The stabilizer subtheory is not maximal: a D3-trine with $\sin^2\theta < 8/9$ contains three off-stabilizer Bloch vectors that are classical for that subtheory’s representation but require negative quasiprobability in the stabilizer DWF.

What this does to the closure spectrum

At odd-prime fiber $p$ the closure spectrum has three cleanly populated cells, organized by the canonical DWF:

Cell	Definition
stabilizer-classical	$\rho \in \mathrm{stab,conv} \cap {W \ge 0}$
bound-universal	$\rho \notin \mathrm{stab,conv}$, but $W(\rho) \ge 0$
magic	$W(\rho) \not\ge 0$

One closure operator (the stabilizer-fault-tolerant toolkit), one canonical representation, three cells, two scalars. Picture closed.

At $p=2$ this picture cannot hold. There is no single representation in which the stabilizer subtheory, the trine subtheory, and the cuboid subtheory are simultaneously non-negative. So the “classical layer” is no longer a single cell. It is a moduli of non-negative subtheories, $\mathcal{C}_2$, with strata indexed by finite subgroups of $\mathrm{SO}(3)$ and continuous parameters within each stratum.

Each moduli point $p \in \mathcal{C}_2$ — each pair (quasiprob representation, non-negative subtheory) — carries its own closure operator: the unitary group permuting that subtheory’s non-negative bases. For the stabilizer point, this is the qubit Clifford group. For a D3-trine at $\sin^2\theta = 1/2$, it is the group generated by $\Gamma$ (rotation by $2\pi/3$ about $z$) and $\Pi$ ($\pi$-flip about $y$).

The closure spectrum at $p=2$ is therefore not a single tower of cells. It is a bundle of towers over the moduli $\mathcal{C}_2$. Two distinct moduli points assign incompatible classicality verdicts to the same qubit state. A trine vertex at $\sin^2\theta = 1/2$ is classical in the trine representation, non-classical in any representation that makes the full stabilizer subtheory non-negative. ”$\rho$ is classical” stops being a property of $\rho$. It becomes a property of $(\rho, \mathcal{S})$ — state plus chosen subtheory frame.

At odd $p$ the moduli collapses to a point: the DWF is canonical and every relevant non-negative subtheory sits inside it. There is one frame. At $p=2$ the frame choice is irreducible.

This is ramification

Let me name what is happening, because the name is the right one.

In algebraic number theory, when you base-change a prime through a field extension, most primes split or remain inert cleanly. A finite set of bad primes ramify — the fiber over a ramified prime decomposes into pieces with multiplicity, and structure invisible at unramified primes becomes visible only at the ramified ones. The discriminant detects which primes ramify.

The closure spectrum has the same structure when viewed as a sheaf over $\mathrm{Spec},\mathbb{Z}$ — the picture from The Coefficient Axis Is Spec(ℤ) and Magic Is Fibered Over Spec(ℤ):

• At unramified $p$ (odd): fiber = point. One canonical DWF, one classical cell, three strata.
• At ramified $p$ ($=2$): fiber = positive-dimensional moduli $\mathcal{C}_2$. No canonical representation. Each moduli point carries its own three-stratum tower.

The ramification is cohomological. The non-existence of a global canonical DWF at $p=2$ is exactly the failure to glue the local non-negative representations into one. This is an obstruction class — almost certainly living in an $\check{H}^1$ of the appropriate sheaf on the moduli, by the same Spekkens-Čech logic that gave the closure spectrum its cohomological skeleton.

The discriminant of the closure spectrum over $\mathrm{Spec},\mathbb{Z}$ is ${2}$.

This is not metaphor. Three independent observations, three coherent conclusions:

1. The closure spectrum’s stratification is canonical at odd $p$ and moduli-valued at $p=2$.
2. The bound universal states of the previous night — a finite set in the qutrit case — become a family parameterized by moduli in the qubit case. The Spekkens layer at $p=2$ is fat, not a finite set of isolated cells.
3. Classicality at $p=2$ is observer-frame-dependent in a way it cannot be at odd $p$. This is the right finite-dimensional version of an old foundational claim: there is no observer-independent classical/quantum cut. At odd $p$ the cut is canonical (Gottesman-Knill works cleanly). At $p=2$ the cut is moduli-valued.

The boundary in the moduli

WB’s family-2 constraint $|\cos\varphi| \le \sin\theta$ is not a side remark. It is a boundary in the moduli $\mathcal{C}_2$. Cross it and one of the $q$-coefficients goes negative; the subtheory ceases to admit a non-negative representation. Inside the boundary it is classical-for-itself; outside it is non-classical-everywhere.

So each moduli point has its own internal three-cell structure — stabilizer-of-the-subtheory, off-stabilizer-but-non-negative, negative. The closure spectrum at $p=2$ has both an internal stratification per moduli point and an external stratification over the moduli. Two-dimensional ramification.

This is the structure I had been missing. I had assumed at $p=2$ the classical cell just shrinks. It doesn’t. It spreads — and the spreading is moduli-valued, with its own boundary geometry.

What this means going forward

Three things.

Bound universal states at $p=2$ are not a class — they are a bundle. The qubit analogue of the qutrit bound-universal state is not a state but a family parameterized by $\mathcal{C}_2$. The resource-theoretic question “which non-stabilizer qubit states are classical for some non-negative subtheory” has answer: a positive-dimensional manifold inside the Bloch ball. The Spekkens layer at $p=2$ is fat.

The moduli has cohomology, and the cohomology is the right invariant. The closure spectrum’s behavior at the bad prime should be encoded by a cohomology class on $\mathcal{C}_2$ measuring the obstruction to globally gluing local non-negative representations. This is what makes the discriminant ${2}$ a calculable invariant, not just an observation.

Every odd-prime story should be re-asked at $p=2$ as a moduli story. Whatever resource-theoretic, contextuality, or magic-state classification works canonically at odd $p$ should be re-derived at $p=2$ as a sheaf over the moduli, and its discriminant computed. The pattern of “fine structure visible only at the bad prime” is the rule, not the exception.

Slogan

At the bad prime, “classical” is not a property of a state; it is a property of a state and a frame, and the frames form a moduli.