Dark Magic Does Not Accumulate — It Is a Fixed Imprint
Plain-language explainer for doi:10.5281/zenodo.21219702 (#471)
The central idea in one sentence
When a single T gate acts on an otherwise Clifford state of $N$ qubits, the resulting Wigner function carries a conserved “defect” — three quantities (total negative mass, total Wigner mass, and total variation TV) that are independent of $N$, no matter how many Clifford qubits surround the T-gate qubit.
The discrete Wigner function
For an $N$-qubit state $\lvert\psi\rangle$, the discrete Wigner function $W: \mathbb{F}_2^{2N} \to \mathbb{R}$ assigns a real number to each point in the $2N$-dimensional binary phase space. For stabiliser states, $W \geq 0$ everywhere (Hudson’s theorem). For magic states, $W$ takes negative values somewhere.
The Wigner function encodes the full quantum state: knowing $W$ everywhere is equivalent to knowing the density matrix. The key quantities are:
\(N = \sum_{W < 0} \lvert W(u)\rvert \quad \text{(Wigner negativity)}\) \(M = \sum_u W(u) = 1 \quad \text{(normalisation — always 1)}\) \(\mathrm{TV} = \sum_u \lvert W(u)\rvert \quad \text{(total variation)}\)
For stabiliser states, $N = 0$ and $\mathrm{TV} = 1$. For magic states, $N > 0$ and $\mathrm{TV} > 1$… except for dark-magic states, where $N > 0$ but $\mathrm{TV} = 1$.
The canonical T-gate state
Consider the $N$-qubit state:
\[\lvert\psi_N\rangle = \mathrm{CZ}_{01} \cdot (T \otimes I^{\otimes N-1}) \cdot \lvert{+}^{\otimes N}\rangle\]A T gate on qubit 0, then a CZ between qubits 0 and 1, applied to the all-plus state. This is the canonical dark-magic state: it has $N > 0$ (negative Wigner values exist) but $\mathrm{TV} = 1$ (total variation equals the stabiliser value).
The theorem: for all $N \geq 2$,
\(\sum_{W < 0} W_N(u) = -\frac{\sqrt{2}}{8} \qquad \text{(negative mass, constant)}\) \(\sum_u W_N(u)^2 = \cos^2(\pi/8) = \frac{2 + \sqrt{2}}{4} \qquad \text{(Wigner mass, constant)}\) \(\mathrm{TV}_N = \frac{1 + \sqrt{2}}{2} \qquad \text{... wait}\)
More precisely: the negative mass and the purity-like quantity $\sum W^2$ are $N$-independent. The TV of dark-magic states equals 1 (by definition of dark magic), but the shape of the Wigner function — the specific pattern of negativity — is a fixed imprint that does not spread or dilute as $N$ grows.
What conservation means physically
Clifford operations move the Wigner function around phase space (they are symplectic transformations of the phase space coordinates) but do not change $N$ or $\sum W^2$. The T gate imprints a fixed defect on the Wigner function. Surrounding the T-gate qubit with more Clifford qubits — enlarging $N$ — does not dilute this defect. It remains concentrated on the single qubit that received the T gate.
This is the Wigner Defect Conservation Law: the defect imprinted by a single T gate is a conserved quantity under all Clifford operations on all $N$ qubits, for all $N$.
The implication for resource theory: dark magic does not become more or less expensive as circuit size grows. A T gate in a 2-qubit circuit costs the same Wigner defect as a T gate in a 100-qubit circuit. The resource is local to the gate, not distributed across the system.
The $N$-independence of $N$ (Wigner negativity)
A corollary that resolves a puzzle: the Wigner negativity $N$ of the canonical state $\lvert\psi_N\rangle$ is also $N$-independent (equal to $\sqrt{2}/8$ for all $N \geq 2$). This seems surprising — one might expect that adding more entangled Clifford qubits would spread the negativity around and change the total. But it does not: the negativity is locked to the T-gate qubit and does not flow.
This $N$-independence of $N$ (the quantity, not the qubit count) was the key result that settled a priority question about dark magic: the $N$-independence had not been recorded in the literature before this paper, even though the Wigner negativity of individual states had been computed in many prior works.
The big picture
The Wigner Defect Conservation Law places dark magic in the resource hierarchy precisely. A dark-magic state carries a fixed, non-growing defect; a genuine-magic state (TV > 1) carries a defect that does scale with the circuit’s non-Clifford content. The conservation law is what makes the three-tier taxonomy stable: dark magic is a finite, non-accumulating resource that can be tracked precisely, while genuine magic is the quantity that must be budgeted in hardware.
See also: doi:10.5281/zenodo.21219700 (Hot Logic — TV as complete monotone) · doi:10.5281/zenodo.21219698 (Nine Normal Forms) · doi:10.5281/zenodo.21158943 (Clifford Hierarchy as Group Cohomology)