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arXiv:2603.20792

A Phase-Space Geometric Measure of Magic in Qubit Systems

Soumyojyoti Dutta, Tushar

quant-phSubmitted Mar 21, 2026• Announced Mar 24, 2026• 0 citations• Revised Sep 10, 2026 (v3)
v1 · Mar 21, 2026v2 · Mar 24, 2026v3 · Sep 10, 2026v4 · Oct 6, 2026v5 · Oct 11, 2026
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Abstract

Magic -- the resource enabling quantum computational advantage beyond stabilizer circuits -- has a clean phase-space characterization in odd prime dimensions that qubits notoriously lack. We study C(rho), the l_1 distance from a state's discrete Wigner function to the stabilizer polytope, and determine its exact geometry. We prove that the single-qubit Wigner l_1 metric has a cuboctahedral unit ball, that max_rho C(rho) = (sqrt(3)-1)/2 for a single qubit, attained precisely at the eight face states, and that C(rho_1 x ... x rho_n) <= prod_i (1 + C(rho_i)) - 1 for single-qubit factors. Together these give the exact tensor powers ((1+sqrt(3))/2)^n - 1 and the exact maximum of C over fully separable n-qubit states, leaving only entangled states open. Because no discrete Wigner function for qubits is Clifford covariant, any such measure is frame dependent, and we determine exactly which of its features are not. For a single qubit we show the valid frames are exactly eight, and that C is identical on all of them, so the maxima above are properties of the state rather than of the representation. For two qubits the conclusion reverses: enumerating all 6144 translation-covariant frames, they split into four equal classes on which the ratio C(rho_Rx)/C(rho_Ry) takes the values 1/2, 1 and 2. Within the Wootters frame we compute that structure exactly: a tetrahedral dichotomy governing when the product bound is saturated, and integer values 1, 2, 1 of the tightness ratio kappa := (Gamma-1)/C against the robustness of magic Gamma for three families in the [[2,1,1]] codespace, whose optimal witnesses are logical Pauli operators. We prove the bound Gamma >= 1 + C/M_n, and show C is not a magic monotone, so asymptotic distillation rates require Gamma.

Comments
17 pages, 8 figures

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Magic -- the resource enabling quantum computational advantage beyond stabilizer circuits -- has a clean phase-space characterization in odd prime dimensions that qubits notoriously lack. We study C(rho), the l_1 distance from a state's discrete Wigner function to the stabilizer polytope, and determine its exact geometry.
We prove that the single-qubit Wigner l_1 metric has a cuboctahedral unit ball, that max_rho C(rho) = (sqrt(3)-1)/2 for a single qubit, attained precisely at the eight face states, and that C(rho_1 x ... x rho_n) <= prod_i (1 + C(rho_i)) - 1 for single-qubit factors. Together these give the exact tensor powers ((1+sqrt(3))/2)^n - 1 and the exact maximum of C over fully separable n-qubit states, leaving only entangled states open.
Because no discrete Wigner function for qubits is Clifford covariant, any such measure is frame dependent, and we determine exactly which of its features are not. For a single qubit we show the valid frames are exactly eight, and that C is identical on all of them, so the maxima above are properties of the state rather than of the representation. For two qubits the conclusion reverses: enumerating all 6144 translation-covariant frames, they split into four equal classes on which the ratio C(rho_Rx)/C(rho_Ry) takes the values 1/2, 1 and 2.
Within the Wootters frame we compute that structure exactly: a tetrahedral dichotomy governing when the product bound is saturated, and integer values 1, 2, 1 of the tightness ratio kappa := (Gamma-1)/C against the robustness of magic Gamma for three families in the [[2,1,1]] codespace, whose optimal witnesses are logical Pauli operators. We prove the bound Gamma >= 1 + C/M_n, and show C is not a magic monotone, so asymptotic distillation rates require Gamma.
Absolutely stabilizer states are those that remain convex mixtures of stabilizer states after conjugation by any unitary. Here we give a characterization of such states for multiple qudits of all prime dimensions by introducing a polytope of their allowed spectra. We illustrate this through the examples of one qubit, two qubits, and one qutrit. In particular, the set of absolutely stabilizer states for a single qubit is a ball inscribed in the stabilizer octahedron, but for higher dimensions the geometry is more complicated. For odd-prime-dimensional qudits, we also give a complete characterization of absolutely Wigner-positive states, i.e., states whose Wigner function remains nonnegative after conjugation by any unitary. In so doing, we show there are absolutely Wigner-positive states that are not absolutely stabilizer, which can be seen as a unitarily invariant version of bound magic. We then study the radii of the largest balls contained in the sets of absolutely stabilizer states and absolutely Wigner-positive states. These radii respectively tell us the lowest possible purity of nonstabilizer and Wigner-negative states. Conversely, we also find the radius of the smallest ball containing the set of absolutely Wigner-positive states, giving a tight purity-based necessary condition thereof. Finally, we apply our methods to magic witnessing and find simple sufficient conditions for determining if an nnn-qubit state is reducible to an mmm-qubit magic state for any m<nm<nm<n.
The apparent absence of a stabilizer-specific phase-space description for qubit systems is not fundamental, but arises from fixing a state-independent discrete Wigner map. Allowing the phase-space representation to adapt to the state through a simple optimization procedure, we prove that every qubit stabilizer state admits an exact delta-function discrete Wigner representation for any fixed phase-space partition. This optimization can be implemented directly at the level of measurement statistics by selecting maximal probabilities across mutually unbiased bases forming the partition. This construction naturally leads to a non-stabilizer witness defined by the optimized value of the Wigner function at the origin. Stabilizer states saturate the universal lower bound of this witness for pure states, whereas a value strictly larger than the bound certifies non-stabilizerness. Although the witness is not a finite-size resource monotone, it becomes sharply concentrated and effectively invariant under random Clifford transformations in the large-qubit limit, suggesting an asymptotically linear growth law for broad families of non-stabilizer states. Our results restore a meaningful geometric phase-space characterization of qubit stabilizer structure and provide an operational diagnostic of non-stabilizerness.
Magic states enable universal, fault-tolerant quantum computation within the stabilizer framework. Their non-stabilizerness supplies the resource needed to bypass the Eastin-Knill theorem while allowing fault-tolerant distillation. Although many measures of magic exist, not every nonzero-magic state is known to be distillable, and many of currently known distillable states are special cases of Clifford-stabilizer states, defined as pure states uniquely stabilized by finite Clifford subgroups. We develop a general framework for group-covariant functionals on Hermitian operators, introducing the notions of GGG-stabilizer spaces, states, and codes for arbitrary finite subgroups G⊂U(H)G \subset \mathrm{U}(\mathcal{H})G⊂U(H). We define analytic families of GGG-covariant functionals and prove that any GGG-invariant pure state is extremal for a broad class of derived functionals, including symmetric, max-type, and Rényi-type functionals, whenever the underlying family is GGG-covariant. This extremality holds for purity-preserving variations orthogonal to the stabilized subspace. Specializing to the Pauli and Clifford groups, we recover the extremality structure of canonical magic measures such as mana, stabilizer Rényi entropies, generalized Rényi entropies, and stabilizer fidelity, and show that Clifford-stabilizer states extremize them. We classify such states for qubits, qutrits, ququints, and two-qubit systems, identifying new candidates for magic distillation. We further propose an inefficient distillation protocol for a two-qubit magic state with stabilizer fidelity exceeding standard benchmarks, and conjecture that SIC-POVM fiducial states are Clifford-stabilizer states.
We prove that, for an nnn-qubit system of dimension d=2nd=2^nd=2n, every state satisfying Tr⁡(ρ2)≤1/(d−a∗)\operatorname{Tr}(ρ^2)\le 1/(d-a_\ast)Tr(ρ2)≤1/(d−a∗​), with a∗=0.458327⋯a_\ast=0.458327\cdotsa∗​=0.458327⋯, lies inside the stabilizer polytope and is therefore magic-free. Combining this result with general geometric properties of high-dimensional polytopes, we establish quantitative estimates for the Hilbert--Schmidt inradius and volume radius of the stabilizer polytope, and use them to characterize the typicality of magic in random induced states obtained by tracing out a kkk-dimensional subsystem from a d×kd\times kd×k-dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension k⋆k_\stark⋆​ is bounded between Ω(d2/log⁡2d)Ω(d^2/\log^2d)Ω(d2/log2d) and O(d2)\mathcal{O}(d^2)O(d2). We further prove that the number of facets of the stabilizer polytope lies between exp⁡[Ω(d2/log⁡2d)]\exp[Ω(d^2/\log^2 d)]exp[Ω(d2/log2d)] and exp⁡[O(d2log⁡2d)]\exp[\mathcal{O}(d^2\log^2 d)]exp[O(d2log2d)] employing a result of Bourgain and Milman in convex geometry, substantially improving upon the previous quasipolynomial lower bound and implying that doubly-exponentially many linear inequalities in the number of qubits are required for an exact description of the magic-free region. Overall, our results reveal the near-extremal geometry of the stabilizer polytope and provide a quantitative foundation for understanding the typicality, robustness, and detectability of magic.
Coherent errors leave stabiliser-code-encoded magic states in superpositions of syndrome sectors. We ask how much magic these coherent errors add, and whether syndrome extraction depletes logical magic. Using Robustness of Magic, we show that, for a unitary error U on a correctable region which supports no stabiliser, the encoded state has the same robustness as the logical state plus one copy of the Choi state of U , in every stabiliser code encoding. Syndrome extraction removes this copy, returning the logical robustness. The removed magic is positive for every non-Clifford error: adding a non-stabiliser state to any state multiplies one plus the robustness by a (tightly- bounded) factor above one. The magic capacity of two-qubit gates Clifford-equivalent to diagonal or single-qubit gates is equal to their Choi-state robustness. We give a linear-algebraic test for when a Clifford error makes syndrome extraction measure a logical operator, and give magic lifetimes of stored single-qubit magic states under Pauli noise. Choosing the plane of the stored state extends these lifetimes about 1.5x on rotated surface codes under low code-capacity depolarising noise. We identify states and channels in odd-prime dimension for which Wigner negativity and robustness show equal lifetimes.
We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored TTT gate magic state. This lowers its asymptotic bound to 2∼0.463t2^{\sim 0.463 t}2∼0.463t for multi-Pauli measurements on ttt magic states, improving over the best previously found bound of 2∼0.468t2^{\sim 0.468 t}2∼0.468t. We numerically demonstrate this reduction. This constructively produces the most efficient strong simulation algorithm of the Clifford+TTT gateset to relative or multiplicative error. We then examine the cost of Pauli measurement in terms of its Gauss sum rank, which is a slight generalization of the stabilizer rank and is a lower bound on its asymptotic scaling. We demonstrate that this lower bound appears to be tight at low ttt-counts, which suggests that the stabilizer rank found at the twelve-qubit state can be lowered further to 2∼0.449t2^{\sim 0.449 t}2∼0.449t and we prove and numerically show that this is the case for single-Pauli measurements. Our construction directly shows how the reduction at 121212 qubits is iteratively based on the reduction obtained at 666, 333, 222, and 111 qubits. This explains why novel reductions are found at tensor factors for these number of qubit primitives, an explanation lacking previously in the literature. Furthermore, in the process we observe an interesting relationship between the T gate magic state's stabilizer rank and decompositions that are Clifford-isomorphic to a computational sub-basis tensored with single-qubit states that produce minimal unique stabilizer state inner products -- the same relationship that allowed for finding minimal numbers of unique Gauss sums in the odd-dimensional qudit Wigner formulation of Pauli measurements.