An application of separable-state nonrepresentability to Grassmann orbitopes and calibrations

Abstract

We apply Fawzi’s established obstruction for normalized complex separable states to the oriented real Grassmann orbitope C₄,₁₂. Two affine hyperplanes select a section real-linearly isomorphic to Sep(3,3), so this mass ball has no finite projected semidefinite lift. A compact first-moment argument then disproves the universal degree-one supporting sum-of-squares assertion formulated by Sanyal, in joint work with Rostalski. It also gives a negative answer in general to Harvey–Lawson’s Question 6.5: some constant comass-one four-form on R¹² has no pointwise or local norm completion. The failing unnormalized functional may have rational coefficients, but no explicit coefficient list or exact comass is supplied. The reduction combines classical Kähler faces, established fermionic projector bodies and the prior tensor-to-wedge lemma with Fawzi’s theorem; worldwide priority of this assembly is not certified.

The idea in plain English

Semidefinite programming describes a convex set through positivity conditions on a finite matrix, possibly with extra variables that are later projected away. Such descriptions can turn geometric optimization into a standard computational problem. This paper shows that one natural geometric body has no exact description of this kind, regardless of matrix size.

The body is built from all oriented four-dimensional planes in twelve-dimensional real space. The key observation is that two linear conditions select a piece that is equivalent to the set of separable states of two three-level quantum systems. Fawzi had already proved that this quantum-state set has no finite semidefinite description.

That obstruction also answers two geometric questions in the negative. A proposed universal sum-of-squares certificate for supporting inequalities fails, and some calibration—a differential form used to certify volume-minimizing geometry—cannot have the norm-completion identity asked for by Harvey and Lawson.

The proof establishes existence. It does not write down the coefficients of the failing calibration or claim that twelve is the smallest possible ambient dimension. The contribution is a reduction linking established results and transferring their obstruction to this particular geometric setting.

The precise result

Let C₄,₁₂ be the convex hull of all unit decomposable real four-vectors in Λ⁴R¹², including both orientations. The paper identifies C₄,₁₂ ∩ {Φ = 1, Q = 0} with normalized complex Sep(3,3) through an injective real-linear map.

Consequently C₄,₁₂ has no finite projected semidefinite lift, even when arbitrary auxiliary variables and affine equalities are allowed. The conclusion concerns every finite lift, not just one chosen relaxation hierarchy.

There exists a real linear functional ℓ, with δ = max ℓ on the unit decomposable four-vectors, for which δ − ℓ is not a sum of squares of affine-linear polynomials modulo the full real vanishing ideal. The coefficients of the unnormalized ℓ can be rational.

There exists a constant closed four-form φ on R¹² of comass one for which φ(p)² + Σψⱼ(p)² = ‖p‖² cannot hold on all decomposable four-vectors for any finite collection of linear four-forms ψⱼ. Failure at each point also rules out a local differential-form completion.

Why the result holds

The classical Wirtinger inequality selects the positively oriented complex two-planes as a face of the real mass ball. A real-linear projector map identifies that entire face with the standard convex hull of decomposable bivector projectors.

Split C⁶ into two orthogonal copies of C³. A zero-trace condition on the two pure exterior blocks forces every positive-weight generator into the mixed block. There, the prior tensor-to-wedge lemma identifies decomposable bivectors exactly with product tensors.

The resulting section is normalized Sep(3,3). Affine sections and linear maps preserve finite semidefinite lifts, so a lift of the mass ball would contradict Fawzi’s theorem.

A first-moment matrix constrained by all degree-two vanishing relations has fixed trace and a compact feasible set. If every supporting inequality had an affine SOS certificate, its projection would equal the mass ball and provide the forbidden lift.

Strict separation supplies a failing support, with rational coefficients before normalization. A norm-completion identity would turn its normalized supporting inequality into an affine SOS, giving the calibration contradiction.

Scope and limitations

The nonrepresentability theorem for normalized complex separable states is Fawzi’s established result. The paper applies that theorem; it does not prove a new quantum-state nonrepresentability theorem.

The two hyperplane equations are imposed inside the oriented mass ball. They alone do not characterize separable states, and Q is only required to be nonnegative on the selected Kähler face.

The failing calibration is existential. No coefficient list, exact comass, explicit separating certificate or rational normalized calibration is supplied. The classical Kähler form used to expose the face is not the failing form.

The counterexample occurs at (k,n) = (4,12). No claim is made that this is the least dimension, that lower-dimensional cases fail, or that earlier experiments in dimensions at most seven were incorrect.

The relevant SOS polynomials have degree at most one, including constants, and equality is taken modulo the full real vanishing ideal. Earlier obstructions for homogeneous quadratic Grassmann cones concern a different representation and do not by themselves establish this oriented mass-ball claim.

Earlier work already supplied the Kähler-face geometry, fermionic projector body, tensor-to-wedge lemma and SOS/completion connection. The claimed addition is their explicit assembly with Fawzi’s obstruction. This bounded comparison does not establish worldwide first priority, formal verification or completed human peer review.

Status and AI use

AI tools were used extensively in developing the argument, checking proofs, reviewing literature and preparing the paper. Internal AI reviews and exact-coordinate checks do not constitute conventional human peer review or formal certification.

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Collaborate on this result

I welcome substantive collaboration on independent verification of the geometric reduction, its relationship to earlier work on calibrations and semidefinite lifts, and preparation for journal review. Expertise in convex algebraic geometry, calibrated geometry or quantum separability would be especially useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). An application of separable-state nonrepresentability to Grassmann orbitopes and calibrations (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23353160

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Background and prior work

• Fawzi, The Set of Separable States has no Finite Semidefinite Representation Except in Dimension 3 × 2: Theorem 1 excludes arbitrary finite projected Hermitian semidefinite representations of normalized complex Sep(3,3). This is the decisive imported obstruction, first public in 2019.
• Sanyal, Grassmann Orbitopes, Oberwolfach Report 13/2015: Printed pages 708–709 state the universal degree-one SOS conjecture, credit joint work with Philipp Rostalski and connect it with Harvey–Lawson’s question.
• Harvey and Lawson, Calibrated geometries: Question 6.5 asks for a local differential-form norm completion on simple vectors. Their classical Kähler examples have such completions; the present existential obstruction concerns other constant four-forms.
• Vrana and Lévay, Special entangled quantum systems and the Freudenthal construction: Section 3.1, equation (25) and its lemma establish the tensor-to-wedge decomposability interface used here in the C³ ⊕ C³ case.
• Grabowski, Kuś and Marmo, Segre maps and entanglement for multipartite systems of indistinguishable particles: Definition 4.2 and Theorem 5.1 give the established fermionic separable-state body as the convex hull of normalized decomposable projectors.
• Sanyal, Sottile and Sturmfels, Orbitopes: Theorem 7.6 shows that the (3,6) Grassmann orbitope is not a spectrahedron. That unprojected obstruction does not exclude all finite projected lifts.
• Bettiol, Kummer and Mendes, Convex algebraic geometry of curvature operators: Corollary 4.2 and Remark 4.3 already exclude finite shadows for cones of nonnegative homogeneous quadratics on Grassmannians. Their representation is Sym²(ΛᵏRⁿ), distinct from the oriented mass ball in ΛᵏRⁿ.