Magic Monotones (Overview)

If magic is a resource, we should be able to measure how much of it a state has. A magic monotone is a function $M(\rho)$ that quantifies non-stabilizerness, in the same way entanglement monotones quantify entanglement. This lesson surveys the main candidates and the axioms they obey.

What makes a function a monotone

In the resource theory of stabilizer computation the free operations are stabilizer operations (Clifford unitaries, Pauli measurements, stabilizer ancillas, partial trace, and classical randomness).[^veitch-resource] A good magic monotone $M$ should satisfy:

Faithfulness on the free set: $M(\rho) = 0$ if and only if $\rho$ is a stabilizer (mixed) state — i.e. $\rho \in \mathrm{STAB}$ .
Monotonicity: $M$ does not increase under any free (stabilizer) operation, $M(\mathcal{E}(\rho)) \le M(\rho)$ for stabilizer $\mathcal{E}$ .
(Often) additivity / sub-additivity under tensor products, so that $n$ copies carry $\sim n$ times the magic.

Distillation can only concentrate magic, never create it, so any monotone gives a hard bound: you cannot distill more magic out than you put in.

Three important measures

Stabilizer rank $\chi$

The stabilizer rank $\chi(|\psi\rangle)$ is the minimum number of stabilizer states needed to write $|\psi\rangle$ as a linear combination:

|\psi\rangle = \sum_{j=1}^{\chi} c_j\,|\phi_j\rangle, \qquad |\phi_j\rangle \text{ stabilizer.}

Stabilizer states have $\chi = 1$ ; magic states have $\chi > 1$ . The rank controls the cost of classically simulating circuits with magic inputs: a circuit with $t$ copies of $|T\rangle$ can be simulated in time scaling with $\chi(|T\rangle^{\otimes t})$ , which grows sub-exponentially in $t$ . This makes $\chi$ the bridge between magic and classical-simulation hardness.[^bravyi-smith-smolin]

Robustness of magic $\mathcal{R}$

The robustness of magic measures how much stabilizer "noise" must be mixed in to wash a state into the stabilizer polytope. Writing $\rho$ as a quasi-probability combination of stabilizer states $\rho = \sum_i x_i\,\sigma_i$ with $\sum_i x_i = 1$ ,

\mathcal{R}(\rho) = \min\Bigl\{\, \textstyle\sum_i |x_i| \ :\ \rho = \sum_i x_i\,\sigma_i,\ \sigma_i \in \mathrm{STAB} \,\Bigr\}.

It equals $1$ exactly on stabilizer states (where all $x_i \ge 0$ ) and exceeds $1$ for magic states; the amount above $1$ measures the "negativity" needed. Robustness is sub-multiplicative and gives operational bounds on distillation rates and simulation cost.[^howard-campbell]

Mana and Wigner negativity

For systems of odd prime dimension, the discrete Wigner function of a stabilizer state is everywhere non-negative. Mana quantifies the total Wigner negativity,

\mathcal{M}(\rho) = \log \sum_{u} \bigl| W_\rho(u) \bigr|,

and is an additive monotone. Negativity of the Wigner function is, in this setting, exactly the resource that enables quantum speedups — a sharp statement of the old intuition that "negativity = quantumness."

Why monotones matter for distillation

Monotones turn vague statements into theorems. Because every monotone is non-increasing under stabilizer operations, the total magic of the distillation outputs can never exceed that of the inputs. This gives fundamental yield bounds: no protocol can output more high-fidelity magic than the raw supply contains, which is why the resource cost of the previous lesson is unavoidable, not merely an artifact of the $15$ -to- $1$ design. Monotones also certify that a candidate state is genuinely useful (nonzero magic) and bound how efficiently it can be converted to the canonical $|T\rangle$ .

The theme to carry forward: stabilizer = free, magic = costly, and monotones are the currency in which that cost is denominated.

[^veitch-resource]: Veitch et al., The resource theory of stabilizer quantum computation, arXiv:1307.7171. [^howard-campbell]: Howard, Campbell, Application of a resource theory for magic states…, arXiv:1609.07488. [^bravyi-smith-smolin]: Bravyi, Smith, Smolin, Trading classical and quantum computational resources, arXiv:1506.01396.