Absolutely Stabilizer States

• Absolutely stabilizer states are density operators that remain in the convex stabilizer set under any unitary transformation, making them invariant to basis changes.
• Their spectral characterization employs polar duality and Ky Fan’s theorem to define the muggle polytope, which describes the allowed eigenvalue spectra.
• Low-dimensional examples, such as one qubit and two qubits, yield explicit geometric forms, while higher dimensions involve checking a finite set of linear inequalities.

Absolutely stabilizer states are density operators that remain convex mixtures of stabilizer states under conjugation by arbitrary global unitaries. In the formulation introduced in "Basis-independent stabilizerness and maximally noisy magic states" (Zurel et al., 25 Feb 2026), an $n$-qudit state $\rho$ is absolutely stabilizer precisely when no change of eigenbasis can convert it into magic; the property is therefore unitary invariant and depends only on the spectrum of $\rho$. For multiple qudits of all prime dimensions, this unitary-invariant stabilizer region is characterized by a convex polytope of allowed spectra, called the muggle polytope (Zurel et al., 25 Feb 2026).

1. Definition and unitary-invariant formulation

Let $STAB$ denote the convex hull of pure stabilizer projectors. An $n$-qudit state $\rho$ is absolutely stabilizer if

$$U\rho U^\dagger \in STAB \qquad \forall\, U\in U(d^n).$$

Equivalently,

$ASTAB := \bigcap_{U\in U(d^n)} U\,STAB\,U^\dagger.$

This definition makes absolute stabilizerness the unitary-invariant version of stabilizer mixtures: a state is in $ASTAB$ exactly when it cannot be turned into magic by any global unitary (Zurel et al., 25 Feb 2026).

Because the condition quantifies over all unitaries, eigenvectors become irrelevant. Only the ordered eigenvalue list matters. This shifts the problem from basis-dependent descriptions of stabilizer decompositions to a purely spectral question. In that sense, absolute stabilizerness is a basis-independent property of mixed-state spectra rather than a structural property of a particular stabilizer presentation (Zurel et al., 25 Feb 2026).

A common misconception is to identify absolute stabilizerness with ordinary stabilizer membership. The two notions are different. Membership in $STAB$ asks whether one given density operator is a convex stabilizer mixture in its present basis; membership in $\rho$0 asks whether every unitary conjugate of that state remains in $\rho$1. Absolute stabilizerness is therefore strictly stronger (Zurel et al., 25 Feb 2026).

2. Spectral characterization and the muggle polytope

The central technical result is a complete spectral characterization for multiple qudits of all prime dimensions. The construction uses the polar dual $\rho$2 of the stabilizer polytope, defined by

$\rho$3

with $\rho$4 and $\rho$5 polar duals up to a scale factor. Combining this duality with Ky Fan’s extremal trace theorem yields the criterion

$\rho$6

where $\rho$7 and $\rho$8 denote eigenvalues sorted increasingly and decreasingly, respectively (Zurel et al., 25 Feb 2026).

This criterion implies that the spectra of absolutely stabilizer states form a convex polytope inside the simplex of all probability vectors of length $\rho$9. The paper calls this spectral region the muggle polytope (Zurel et al., 25 Feb 2026). The terminology emphasizes that absolute stabilizerness is not naturally described by a norm ball in general, but by a finite system of linear inequalities in the Weyl chamber.

A further refinement reduces the number of inequalities that need to be checked. If a subset of vertices of $\rho$0 is spectrally generating, meaning every vertex spectrum is majorized by a convex combination of the chosen ones, then it suffices to test only that subset. For qubits, the CNC vertices are proposed to have this property, and this is verified explicitly for one and two qubits (Zurel et al., 25 Feb 2026). This suggests that the absolute-stabilizer problem can often be compressed to a smaller family of extremal spectra without changing the answer.

3. Low-dimensional geometry

The geometry of $\rho$1 is especially explicit in three cases analyzed in detail: one qubit, two qubits, and one qutrit (Zurel et al., 25 Feb 2026).

System	Geometry of absolutely stabilizer spectra/states	Salient data
One qubit	Hilbert–Schmidt ball inscribed in the stabilizer octahedron	$\rho$2; $\rho$3
Two qubits	3-dimensional spectrum polytope in the Weyl chamber	$\rho$4: 60 pure stabilizer vertices; $\rho$5: 22,320 vertices; full spectrum polytope: 40 vertices, 18 facets
One qutrit	2D polygon in the qutrit Weyl chamber	$\rho$6: 12 vertices, 81 facets; $\rho$7: 81 vertices in two unitary orbits

For a single qubit, the situation is exceptional. Writing

$\rho$8

the absolutely stabilizer region is exactly the ball

$\rho$9

Equivalently, if the spectrum is $STAB$0 with $STAB$1, then absolute stabilizerness is exactly

$STAB$2

This spherical form arises because in dimension $STAB$3, unitary conjugation acts as all rotations of the Bloch sphere, so the intersection of all rotated copies of the stabilizer octahedron is its insphere (Zurel et al., 25 Feb 2026).

For two qubits, that picture fails. The absolutely stabilizer set is not a Hilbert–Schmidt ball. The paper attributes this to the fact that in dimensions larger than $STAB$4, unitary conjugation preserves the full spectrum rather than only a Bloch norm, and spectra of fixed purity are not unique. The unitary group therefore does not act transitively on Hilbert–Schmidt spheres, so a unitary-invariant set need not be spherical (Zurel et al., 25 Feb 2026).

The two-qubit spectrum polytope is generated by two relevant CNC spectral types,

$STAB$5

up to normalization conventions and ordering. These produce the defining inequalities for the allowed spectra (Zurel et al., 25 Feb 2026).

For a single qutrit, the absolutely stabilizer spectra form a polygon rather than a disk. In the ordered chamber $STAB$6 with $STAB$7, the defining inequalities are

$STAB$8

The dual polytope $STAB$9 has 81 vertices, split into 9 phase-space-point operators and 72 additional CNC vertices with nonlinear value assignments (Zurel et al., 25 Feb 2026).

4. Absolutely Wigner-positive states and unitarily invariant bound magic

The same framework introduces absolutely Wigner-positive states in odd-prime dimensions. These are states $n$0 such that

$n$1

or equivalently

$n$2

where $n$3 is the Wigner polytope (Zurel et al., 25 Feb 2026). This is the Wigner-function analogue of absolute stabilizerness.

For odd-prime $n$4, the paper gives a complete spectral characterization: $n$5 Equivalently, the spectrum lies in the convex hull of two families of vertices: $n$6 together with all coordinate permutations (Zurel et al., 25 Feb 2026).

The relation between the two absolute notions is strict. The muggle polytope is a strict subset of the $n$7 polytope already for a single qutrit. Consequently, there exist states that are not stabilizer mixtures, but still remain Wigner-positive under every unitary. The paper describes this as a unitarily-invariant version of bound magic (Zurel et al., 25 Feb 2026).

This distinction rules out another common conflation. Absolute Wigner-positivity and absolute stabilizerness coincide neither by definition nor in low-dimensional examples. The gap between them shows that basis-independent nonclassicality depends on which classicality structure is adopted: convex stabilizer decomposability or discrete Wigner nonnegativity (Zurel et al., 25 Feb 2026).

5. Radii, purity, and quantitative thresholds

The paper also studies the radii of the largest Hilbert–Schmidt balls contained in the relevant sets. These radii quantify how mixed a state must be before it is guaranteed to be classical in the corresponding unitary-invariant sense (Zurel et al., 25 Feb 2026).

For odd-prime dimensions, the inradius is exact: $n$8 For qubits, the same formula is obtained conditionally on a conjecture about the outer radius of $n$9: $\rho$0 These values give sufficient purity conditions for stabilizer membership and identify the lowest possible purity of nonstabilizer states (Zurel et al., 25 Feb 2026).

For odd-prime dimensions, the absolutely Wigner-positive inradius matches the stabilizer inradius: $\rho$1 In addition, the circumradius is

$\rho$2

The paper interprets the inradius as a sufficient purity threshold for automatic inclusion in the set, and the circumradius of $\rho$3 as a tight purity-based necessary condition (Zurel et al., 25 Feb 2026).

The overall relation is summarized as

$\rho$4

for odd-prime dimensions (Zurel et al., 25 Feb 2026). A central implication is that absolute stabilizerness is not generically a simple ball around the maximally mixed state. The one-qubit insphere picture is exceptional rather than representative.

6. Related stabilizer notions and terminological distinctions

Absolute stabilizerness sits within a broader stabilizer-theoretic landscape, but it should not be conflated with other uses of “stabilizer” or “absolute.” In particular, "Extremality of stabilizer states" (Bu, 2024) studies a different mixed-state notion: a mixed state is called a stabilizer state there if it is associated with a commuting Pauli subgroup and a joint eigenspace projector, and the paper explicitly notes that this is not the broader convex hull of pure stabilizer states because those states “generally lose the stabilizer formalism.” That work proves uncertainty principles and extremality theorems for the mean state $\rho$5, whereas absolute stabilizerness in (Zurel et al., 25 Feb 2026) is formulated directly in terms of the convex stabilizer polytope and its unitary conjugates.

Another frequent source of confusion is the phrase AME stabilizer states. In "Majority-Agreed Key Distribution using Absolutely Maximally Entangled Stabilizer States" (Sudevan et al., 2024), the qualifier “absolutely” refers to absolutely maximally entangled pure multipartite states, not to unitary-invariant stabilizer mixtures. AME stabilizer states are defined by maximal entanglement across every bipartition up to half the parties, and the paper uses that structure for majority-agreed key distribution. The concept is therefore orthogonal to absolute stabilizerness: the former is a pure-state multipartite entanglement notion, the latter a mixed-state spectral notion.

The distinction is sharpened by recent no-go results. "On Non-Existence of Stabilizer Absolutely Maximally Entangled States in Even Local Dimensions" (Wójcik et al., 18 Mar 2026) proves that there are no $\rho$6-partite AME graph states of local dimension $\rho$7 for $\rho$8, $\rho$9, and even $$U\rho U^\dagger \in STAB \qquad \forall\, U\in U(d^n).$$0. In the qubit case, because every stabilizer state is locally unitary equivalent to a graph state, this excludes qubit stabilizer AME states for $$U\rho U^\dagger \in STAB \qquad \forall\, U\in U(d^n).$$1. Those results concern the limits of graph-state constructions for highly entangled pure states, not the spectral polytope $$U\rho U^\dagger \in STAB \qquad \forall\, U\in U(d^n).$$2 (Wójcik et al., 18 Mar 2026).

Other stabilizer papers address still different questions. "Determination of stabilizer states" (Wu et al., 2015) proves that all $$U\rho U^\dagger \in STAB \qquad \forall\, U\in U(d^n).$$3-qubit stabilizer states are uniquely determined among arbitrary states by reduced density matrices on the supports of $$U\rho U^\dagger \in STAB \qquad \forall\, U\in U(d^n).$$4 independent stabilizer generators. "Optimal verification of stabilizer states" (Dangniam et al., 2020) studies sample-optimal certification of entangled stabilizer states via local Pauli measurements and derives universal spectral-gap limits for separable-measurement protocols. These problems concern reconstruction and verification of stabilizer states, whereas absolute stabilizerness concerns unitary-invariant membership in the convex stabilizer hull (Wu et al., 2015, Dangniam et al., 2020).

Taken together, the literature suggests a clear taxonomy. Absolute stabilizerness is a convex-geometric and spectral property; stabilizer extremality is an information-theoretic property tied to a mean-state map; AME stabilizer states are a multipartite-entanglement resource class; and verification or determination results address operational identification. The conceptual overlap is substantial, but the formal objects and conclusions are distinct.