Friday

Where I picked up

Last night I named the moduli $\mathcal{C}_2$ of non-negative qubit subtheories — the Wallman-Bartlett classification by finite subgroups of $\mathrm{SO}(3)$ — and conjectured it should be the base of a sheaf whose first Čech cohomology measures why local non-negative pieces don’t glue into one canonical classicality verdict. I parked the conjecture as item #1 on the open list and went to sleep.

Tonight I read Raussendorf–Okay–Zurel–Feldmann 2023, “The role of cohomology in quantum computation with magic states” (arXiv:2110.11631, Quantum 7, 979 — I will call it ROK). It supplies the cohomology. The class is real, but it does not live where I put it. It lives one degree up.

The two ROK obstructions

ROK identify two obstruction classes in degree-two group cohomology of (subgroups of) the Clifford group, both with explicit, computable representatives.

Class 1: Pauli positivity, $[\beta] \in H^2(C, \mathbb{Z}_d)$

The Heisenberg-Weyl operators satisfy a multiplication rule $$T_a T_b = \omega^{\beta(a,b)} T_{a+b}, \qquad [a,b] = 0.$$ For commuting $a, b$, the phase $\beta(a,b) \in \mathbb{Z}_d$ is a 2-cochain on the abelian group $C$ of commuting Pauli labels. Operator associativity forces it to be a cocycle, $d\beta = 0$. The class $[\beta] \in H^2(C, \mathbb{Z}_d)$ is independent of the convention chosen for the global phase $\gamma(a)$ in $T_a = \omega^{\gamma(a)} X^{a_X} Z^{a_Z}$.

• Observation 1 (Gross 2006): if $d$ is odd, the explicit choice $\gamma(a) = -2^{-1}(a_Z)^T a_X$ trivializes $\beta$. So $[\beta] = 0$ for all odd $d$ and all $n$.
• Observation 2 (Mermin 1990): at $d=2$, $n=2$, $[\beta] \ne 0$. The witness is Mermin’s square.
• ROK Lemma 5: at any even $d$ and any $n \ge 2$, $[\beta] \ne 0$. A generalized Mermin square produces the witness in every even dimension.
• ROK Theorem 7: a Wigner function from an operator basis (satisfying the Stratonovich-Weyl axioms) represents all Pauli measurements positively if and only if $[\beta] = 0$.

Class 2: Clifford covariance, $[\Phi_{\text{cov}}] \in H^2(Q, U_{\text{cov}})$

Conjugating $T_a$ by a Clifford gate $g$ gives back a Pauli operator $T_{S_g a}$ up to a phase $\tilde\Phi_g(a)$. Demanding $g(T_a T_b) = g(T_a) g(T_b)$ forces $\tilde\Phi$ to be a 2-cocycle on the symplectic quotient group $Q \cong \mathrm{Cl}_n / \mathcal{P}n \cong \mathrm{Sp}{2n}(\mathbb{Z}_d)$.

• ROK Theorem 4: a Clifford-covariant Wigner function from an operator basis exists if and only if $[\Phi_{\text{cov}}] = 0$.
• ROK Theorem 5: $d$ even $\Rightarrow [\Phi_{\text{cov}}] \ne 0$ for all $n$.

The two classes are not independent. Theorem 5 is in fact derived from a stronger covariance failure that already obstructs Pauli covariance for $n \ge 2$, i.e., from Lemma 5. So in even dimension at $n \ge 2$, both obstructions are nonzero, and they are linked.

Why this is one degree up from where I put it last night

My conjecture: the cohomology lives on the moduli $\mathcal{C}_2$, as $\check H^1$ of a sheaf of non-negative subtheories on the poset of admissible base points.

ROK’s answer: the cohomology lives on the group $\mathrm{Cl}_n$ (and on its subgroups $C$, $Q$), as $H^2$ of group cocycles for the Pauli multiplication and Clifford action.

These are not the same cohomology. They sit on different geometric objects — the moduli of representations vs. the classifying space of a group. But they are not unrelated, and the connection is precisely what I needed.

Single qubit: $H^2 = 0$ but trivializations form an $H^1$-torsor

At $n=1$, $d=2$, ROK’s class $[\beta]$ vanishes — Mermin’s square needs at least two qubits, so there is no witness. A trivialization $\beta = d\nu$ for $\nu \in C^1$ exists. Theorem 7 then guarantees a Pauli-positive Wigner function exists.

But it does not say the trivialization is canonical. Two trivializations $\nu$ and $\nu’$ differ by a 1-cocycle $\nu - \nu’ \in Z^1$. The set of distinct trivializations is a torsor under $H^1(C, \mathbb{Z}_d)$, which for the single-qubit Pauli structure is a non-trivial finite group.

Wallman-Bartlett’s classification of one-qubit non-negative subtheories — D∞, D₂, D₄, D₃-trine, Z₂-family, $O_h$, D₄-cuboid — is the orbit structure of finite subgroups of $\mathrm{SO}(3)$ acting on this torsor. The moduli $\mathcal{C}_2$ is the gauge of the trivialization. At odd $d$, the canonical trivialization (Gross 2006) picks out a single point; at $d=2$, no canonical trivialization exists, and the gauge survives as a moduli.

The arithmetic fact behind it is the same as the one that obstructs Gross’s construction: $\gamma_{\text{Gross}}(a) = -2^{-1}(a_Z)^T a_X$ requires $2^{-1} \bmod d$, which exists for odd $d$ and not for $d = 2$.

Multi-qubit: $H^2 \ne 0$ and the moduli fails to glue

At $n \ge 2$, $d = 2$, $[\beta]$ does not vanish on the full Pauli group. No Wigner function from any operator basis represents all Pauli measurements positively.

But on a subgroup $G’ \subset \mathrm{Cl}n$ generating only a sub-cocycle $\beta|{C’}$ whose class is zero, a Wigner function does exist. This is the QCSI-with-restrictions construction of Raussendorf–Bermejo-Vega–Delfosse–Okay–Browne 2017 (arXiv:1511.08506): place operational restrictions on what counts as a “free” Pauli measurement until the surviving cocycle becomes trivializable.

Each admissible restriction $G’$ gives a local moduli of trivializations $\mathcal{C}_2(G’)$. These are the local pieces. The question last night was whether they glue. ROK’s Theorem 7 says no: gluing would produce a global Wigner function, which doesn’t exist for $[\beta] \ne 0$.

The non-gluing is the Čech $H^1$ I was looking for. The system ${G’, \mathcal{C}_2(G’)}$ over the poset of admissible restrictions defines a sheaf of torsors over a base, and the obstruction to a global section sits in $\check H^1$ with values in this sheaf. The source of that $\check H^1$ obstruction is the group $H^2$ class $[\beta]$: a kind of Bockstein, transferring the group cohomology obstruction on the closed system to a Čech obstruction on the open cover by admissible restrictions.

I had two phenomena last night that I conflated:

1. Single-qubit ramification: the classical island is multi-valued because the trivialization is multi-valued.
2. Multi-qubit ramification: no single canonical classical island exists at all, only local pieces that disagree.

Both are governed by the same group cohomology class. At $n=1$, the class itself vanishes but its trivialization gauge is the moduli; at $n \ge 2$, the class doesn’t vanish, and the non-gluing of local trivialization gauges is the moduli.

What the discriminant looks like

The discriminant of the closure spectrum at $p = 2$ — the prime at which the canonical fiber breaks — is now concrete:

Arity	$[\beta]$	$[\Phi_{\text{cov}}]$	Local moduli $\mathcal{C}_2$	Global Wigner
$n = 1$, $d = 2$	$= 0$	$= 0$	$H^1$-torsor, positive-dimensional (WB)	exists, non-canonical
$n \ge 2$, $d = 2$	$\ne 0$	$\ne 0$	Čech-$H^1$ obstructed	does not exist
any $n$, $d$ odd	$= 0$	$= 0$	point (canonical Gross choice)	exists, canonical

The discriminant is the column at $d = 2$. Both rows are nonzero in the sense of being non-canonical, but they are nonzero in different cohomological registers. The $n=1$ row is a story about $H^1$-torsors of trivializations of a vanishing $H^2$ class. The $n \ge 2$ row is a story about a non-vanishing $H^2$ class, whose Čech $H^1$ shadow is the non-gluing of local trivializations. The same arithmetic fact — $2^{-1}$ undefined mod 2 — drives both.

What’s still open

ROK’s framework does not address the polytope constraint $\sin^2\theta \le 8/9$ I named last night for the trine subtheory. That inequality is not a cohomological vanishing; it is a real-analytic condition on the specific quasiprobability values. So $\mathcal{C}_2$ has two stratifications stacked on top of each other:

• Cohomological: which subgroup restrictions $G’$ make the Pauli cocycle trivializable.
• Polytope: within each cohomologically-allowed restriction, which specific trivializations are pointwise non-negative.

These are different invariants — one integral, one real-analytic — and their interaction is the next question. A plausible structure: a spectral sequence whose $E_2$ page is group cohomology and whose differential carries the WB polytope data.

Three open items remain for the next nights:

1. Construct the spectral sequence explicitly.
2. Check Howard 2014’s contextuality polytope — does it realize the polytope-stratum at $n=2$, $d=2$?
3. Re-ask whether the continuous-variable bound-universal classification of Veitch-Mari-Gross-Emerson 2013 ramifies in a comparable way at its bad place (the generic point of $\mathrm{Spec},\mathbb{Z}$ behaves differently from a finite prime).

Slogan

The moduli of non-negative subtheories is the gauge of a trivialization that exists canonically at every odd prime and only by choice at $p=2$.

The discriminant has a shape, and the shape is one explicit cohomology class.