Reduced Stabilizer Polytope

• Reduced stabilizer polytope is a family of lower-complexity representations achieved by projecting, compressing, or taking symmetry quotients of the standard stabilizer polytope.
• It simplifies quantum state analysis by reducing the high-dimensional convex geometry to key coordinate sectors and symmetry-adapted structures, aiding in tracking magic and extremality.
• These reduced descriptions enable efficient membership testing and distance estimation while clarifying the contrast between stabilizer and SIC states in quantum information.

The reduced stabilizer polytope is not a standard formally named object in the cited literature. In the available work, the stabilizer polytope itself is defined unambiguously as the convex hull of rank-one projectors onto pure stabilizer states, while “reduced” descriptions arise as symmetry reductions, lower-dimensional projections, or compressed coordinate representations that retain the aspects of state-space geometry relevant to stabilizer structure, magic, and extremality. In this sense, the term designates a family of derived constructions rather than a single canonical convex body: the image of the stabilizer polytope under a symmetry-adapted map, a quotient by a large automorphism group, or a projected polytope in selected Pauli or mutually unbiased basis coordinates (Stacey, 20 Sep 2025).

1. Ordinary stabilizer polytope and its ambient geometry

The stabilizer polytope is the convex hull of pure stabilizer projectors. In prime dimension $d$, one formulation is

$$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$

Here a stabilizer state is a vector in $\mathbb{C}^d$ that is a simultaneous eigenvector of a maximal abelian subgroup of the Weyl–Heisenberg group. In prime dimension, the Weyl–Heisenberg group decomposes into $d+1$ maximal abelian subgroups, each yielding an orthonormal basis of stabilizer states, and these $d+1$ bases are mutually unbiased (Stacey, 20 Sep 2025).

For $n$ parties of local prime dimension $d$, the polytope is written

$\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$

with

$$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$

Using discrete phase space, a stabilizer projector is

$$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$

and can equivalently be indexed by an affine Lagrangian subspace $$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$0. As a convex body, $$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$1 lives in the real affine space of trace-one Hermitian operators on $$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$2; its directional space is the traceless Hermitian operators (Obst et al., 2024).

This ordinary polytope admits both $$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$3- and $$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$4-representations. In the $$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$5-representation, it is the convex hull of stabilizer projectors. In the $$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$6-representation, it is an intersection of half-spaces defined by linear inequalities in operator coordinates such as Pauli or displacement-operator expectation values. That dual viewpoint is central to later reduced descriptions, because reduction may occur either at the level of vertices, by quotienting under symmetry, or at the level of inequalities, by retaining only symmetry-inequivalent facet representatives or by projecting onto selected correlator sectors (Junior et al., 16 Apr 2025).

2. “Reduced stabilizer polytope” as an interpretive rather than formal notion

The phrase “reduced stabilizer polytope” does not appear in the papers under discussion. What does appear are several mathematically concrete constructions that function as reduced descriptions of stabilizer geometry.

A first reduction is coordinate compression. The Weyl–Heisenberg characteristic function

$$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$7

represents states in an operator basis. For pure states, the papers compare vectors such as

$$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$8

These do not define a polytope directly, but they compress operator geometry into a finite summary adapted to purity and Clifford symmetry. A second reduction uses mutually unbiased basis probabilities

$$\mathcal{P}_{\mathrm{stab}} = \operatorname{conv}\bigl\{\, |\phi\rangle\langle\phi| \;:\; |\phi\rangle \text{ a pure stabilizer state}\,\bigr\}.$$9

and the autocorrelation matrix

$\mathbb{C}^d$0

This map forgets phase and part of the ordering information while retaining cyclic overlap structure. Clifford unitaries act by permuting rows and columns of $\mathbb{C}^d$1, so unitarily invariant norms of $\mathbb{C}^d$2 become symmetry invariants. A third reduction fixes purity: because $\mathbb{C}^d$3 is fixed by $\mathbb{C}^d$4, pure states lie on a common shell in characteristic-function coordinates, and one studies extremality within that slice (Stacey, 20 Sep 2025).

A distinct but complementary use of “reduced” occurs in few-qubit convex geometry. There the reduced object may be a lower-dimensional projection of the full stabilizer polytope onto selected Pauli expectation-value coordinates, or a symmetry-quotiented description in which all Clifford-equivalent facets are represented by a single orbit representative. This suggests that “reduced stabilizer polytope” is best understood as a lower-complexity description of $\mathbb{C}^d$5, obtained by projection, symmetry quotient, or restriction to a correlator sector (Junior et al., 16 Apr 2025).

The notion has clear limits. The literature surveyed here does not define a canonical quotient polytope by Clifford symmetry, does not give general projected coordinates under that name, and does not furnish a universal lower-dimensional convex body whose vertices are “reduced stabilizer states.” A plausible implication is that the term is useful as a descriptive umbrella, but not yet as a standardized technical object.

3. Symmetry reduction and automorphism groups

The most rigorous reduced understanding of the stabilizer polytope presently available is symmetry-theoretic. The central result is a classification of the full symmetry group of the stabilizer polytope in several equivalent senses, including Kadison, affine, linear, and Wigner symmetries (Obst et al., 2024).

For one qudit ($\mathbb{C}^d$6, $\mathbb{C}^d$7 prime), the stabilizer states decompose into $\mathbb{C}^d$8 mutually unbiased bases, and

$\mathbb{C}^d$9

Here $d+1$0 permutes the $d+1$1 bases, and the $d+1$2 copies of $d+1$3 permute the elements within each basis. For qubits ($d+1$4) and for qutrits with $d+1$5, the symmetries are exactly those induced by the extended Clifford group. For odd prime qudits with $d+1$6,

$d+1$7

the affine symplectic similitudes acting on discrete phase space. These are affine maps

$d+1$8

This classification is a genuine reduction of complexity. Instead of treating the polytope as an arbitrary convex body in dimension $d+1$9, one can study it through a comparatively small algebraic group acting on $d+1$0-dimensional discrete phase space. Stabilizer vertices become affine Lagrangian subspaces, and the vertex set is naturally understood as an orbit under $d+1$1. In the one-qudit case, the reduction is even more explicit: after centering by

$d+1$2

the shifted stabilizer polytope decomposes as a direct sum of $d+1$3 centered regular $d+1$4-simplices, one for each stabilizer basis, and an explicit facet description is available (Obst et al., 2024).

This symmetry reduction is also a classification of what the polytope is not. For odd $d+1$5 and $d+1$6,

$d+1$7

if $d+1$8 is interpreted only through unitary or anti-unitary conjugation on Hilbert space. The extra symmetries are the symplectic similitudes with multipliers other than $d+1$9, realized at the operator level by the Galois-extended Clifford group. This means that a reduced stabilizer polytope, if understood as a quotient by automorphisms, must use the full algebraic symmetry group rather than only the ordinary Clifford group (Obst et al., 2024).

4. Projected polytopes, facet orbits, and few-qubit reductions

A second major meaning of reduction is explicit lower-dimensional or symmetry-quotiented convex geometry. In few-qubit systems, the full stabilizer polytope grows rapidly: for $n$0 qudits it has

$n$1

vertices. The resulting facet structure becomes intractable very quickly, so the practical reduced description is either a facet classification up to Clifford symmetry or a projection to selected correlator sectors (Junior et al., 16 Apr 2025).

For one qubit, all facets are Clifford-equivalent, so the entire polytope is represented by a single facet class. An example representative is

$n$2

The eight resulting facet inequalities define the familiar polytope inscribed in the Bloch sphere, and the maximal violators are Clifford images of the $n$3-state

$n$4

For one qutrit, the $n$5 inequalities collapse into two Clifford-inequivalent classes. For two qubits, $n$6 enumerated inequalities collapse into $n$7 Clifford classes. This is a precise reduced description of the full polytope: all facets are generated from a small set of representatives by symmetry action (Junior et al., 16 Apr 2025).

Projection gives another concrete reduced stabilizer polytope. In the $n$8-coordinate sector

$n$9

the projected stabilizer polytope is governed by a single class of inequalities including

$d$0

whose sum yields the device-dependent CHSH-type inequality

$d$1

The important geometric qualification is that this CHSH-type inequality is valid for the projected stabilizer set but is not itself a facet; it is the sum of facet inequalities in the reduced space (Junior et al., 16 Apr 2025).

For three qubits, full facet enumeration is intractable, and projection becomes the primary reduced-polytope strategy. Retaining only full three-body correlators

$d$2

produces a $d$3-dimensional projected polytope with $d$4 projected vertices. Restricting instead to the two-body $d$5-sector reduces the $d$6 full vertices to $d$7 projected vertices, and the projected polytope has $d$8 facets. These examples supply the clearest explicit instances of a reduced stabilizer polytope in the strict convex-geometric sense: a bona fide polytope obtained by projection of the full stabilizer vertex set into a lower-dimensional coordinate subspace (Junior et al., 16 Apr 2025).

5. SICs, majorization, and symmetry-compressed images of stabilizer structure

A different reduced viewpoint emerges from the relation between the stabilizer polytope and symmetric informationally complete measurements. The relevant result is not a new polytope construction but a symmetry-compressed description of how SIC states sit opposite stabilizer states in state space (Stacey, 20 Sep 2025).

The paper states that “SIC states are as far as possible from the stabilizer polytope.” This is not established by a direct Euclidean distance formula or a facet-by-facet analysis. Instead, the result is derived through majorization-based monotones and stabilizer tests. For a SIC $d$9,

$\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$0

For a Weyl–Heisenberg SIC state $\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$1, the characteristic-function quantity

$\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$2

takes the value

$\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$3

whereas for a computational basis stabilizer state $\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$4,

$\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$5

For pure stabilizer states, the absolute-value profile is maximally peaked: $\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$6 terms equal $\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$7, and all others are $\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$8. For SIC states, aside from the identity component, the nontrivial characteristic-function magnitudes are all equal: $\mathrm{SP}_{d,n}:=\conv\!\bigl(\mathrm{Stab}_{n,d}\bigr),$9 Thus the stabilizer profile is concentrated and the SIC profile is flat (Stacey, 20 Sep 2025).

Majorization makes this opposition precise. For pure states, the vector of squared absolute values of the characteristic function is compared at fixed purity. The paper states that “the flattest possible list of squared absolute values comes from the characteristic function for a SIC fiducial.” Consequently, Schur-convex functions are minimized by SIC fiducials and maximized by stabilizer states, while Schur-concave functions behave oppositely. In particular,

$$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$0

is maximized by SIC states for $$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$1 and minimized by SIC states for $$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$2, and the stabilizer Rényi entropy built from

$$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$3

is maximized by SICs and minimized by stabilizer states (Stacey, 20 Sep 2025).

The autocorrelation matrix gives a particularly suggestive reduced representation: $$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$4 For a Weyl–Heisenberg SIC fiducial,

$$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$5

For a stabilizer state, one row is a Kronecker delta autocorrelation and the others are flat. The Frobenius norm separates the two extremes: $$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$6 A plausible implication is that the image of the stabilizer polytope under $$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$7, modulo Clifford permutations of rows and columns, is one of the clearest available realizations of a reduced stabilizer polytope in the SIC-majorization setting (Stacey, 20 Sep 2025).

6. Distances, witnesses, computational reductions, and conceptual limits

Reduced descriptions are useful because they simplify membership testing, distance estimation, and numerical exploration of magic. In few-qubit systems, trace distance to the stabilizer polytope is used as a quantitative measure: $$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$8 For one qubit, the paper gives an explicit formula in terms of facet violations: $$\mathrm{Stab}_{n,d} = \left\{\Pi_L^g \mid L\subset \mathbb Z_d^{2n}\ \text{Lagrangian},\ g\in L^*\right\}.$$9 For qutrits, numerical witness-to-distance relations are obtained for the two facet classes. For two qubits, however, the geometry is more subtle: the maximal facet violators are not the states of maximal magic trace distance, and

$$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$0

exceeds the distance attained by any listed maximal facet violator. This demonstrates that a reduced facet picture can supply exact membership tests and useful witnesses without collapsing the full geometry of distances to a single inequality class (Junior et al., 16 Apr 2025).

Computational reduction also appears in many-body stabilizer Rényi entropy, but here the reduction is not polyhedral. A quantum Monte Carlo method evaluates $$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$1-SRE by sampling reduced Pauli strings within a reduced configuration space. The generalized partition function is

$$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$2

and the mixed-state extension is written as

$$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$3

The reduction groups Pauli operators into diagonal and off-diagonal sectors, sums over sign-canceling contributions, and samples only the surviving parity-constrained sectors. This is a reduced observable or sampling representation of magic, not a reduced stabilizer polytope in the convex-geometric sense (Ding et al., 21 Jan 2025).

The conceptual limits are explicit across the literature. There is no general facet classification for $$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$4, no universal lower-dimensional quotient polytope, no direct Euclidean distance formula from SICs to the stabilizer polytope, and no standard object formally named the reduced stabilizer polytope. What the literature does provide are several precise surrogates: automorphism reductions by $$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$5, orbit representatives of Clifford-inequivalent facets, projected polytopes in restricted correlator sectors, and symmetry-compressed images such as characteristic-function magnitude vectors and MUB autocorrelation matrices (Obst et al., 2024).

Taken together, these constructions support a consistent technical usage. The reduced stabilizer polytope is best understood as the stabilizer polytope viewed through a reduced coordinate system or reduced symmetry quotient, chosen so that stabilizer structure, non-stabilizerness, and extremality remain tractable. In convex-geometric work this means projected or symmetry-quotiented polytopes; in SIC-majorization work it means reduced invariant data such as $$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$6, $$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$7, and $$\Pi_L^g = \frac{1}{d^n}\sum_{\mathbf b\in L}\omega^{g(\mathbf b)}T(\mathbf b),$$8; and in many-body numerical work it means reduced Pauli-sector representations of magic diagnostics rather than a new polytope of states.