Minimum Bell-Setting Complexity for Qubit POVM–PVM Separation

Abstract

For every bipartite Bell architecture with two inputs per party and arbitrary finite, input-dependent output alphabets, the shared-randomness convex hull of fixed-qubit POVM behaviors equals the corresponding convex hull of fixed-qubit PVM behaviors. The paper also gives an explicit rational \(3\times2\) Bell functional, an exact qubit-POVM strategy, and a global analytic upper bound for every fixed-qubit PVM strategy with a strict exact gap. Hence, up to exchanging the parties, three inputs versus two are necessary and sufficient for a linear Bell-functional separation.

1. Main theorem

Let \(\overline{\mathcal Q}^{\mathrm{POVM}}_2(\mathbf A,\mathbf B)\) and \(\overline{\mathcal Q}^{\mathrm{PVM}}_2(\mathbf A,\mathbf B)\) denote the shared-randomness convex hulls of behaviors generated on local Hilbert spaces of dimension at most two. Output sets may be finite and depend on the input. Zero projectors, deterministic relabelings, and stochastic output postprocessing are permitted; dimension-increasing Naimark dilations are not.

With one input on either party every nonsignaling behavior is local. Combining that fact, the universal two-input equality, and the exact \(3\times2\) witness proves the minimum-setting classification.

2. Proof mechanism

One binary party

When one party is binary on both inputs, every behavior admits one common shared-randomness decomposition into fixed-qubit PVM behaviors. The proof compresses steered qubit operators into a three-dimensional Lorentz cone, decomposes positive relations into two-versus-two circuits, and lifts each circuit to two projective decompositions of one qubit state.

Only one residual architecture

An extreme hypothetical separator reduces to a pure entangled state with one binary PVM and one extremal ternary rank-one POVM on each party. A common-span perturbation and the extremal qubit rank-square bound exclude four-outcome POVMs and all other architectures.

Lorentz incidence and rank trichotomy

The remaining strategy becomes a physically exact fourteen-dimensional Lorentz-incidence manifold. Finite POVM duality forces strictly positive determinant multipliers at a strict separator. The weighted incidence second form has ambient inertia \((4,12)\). The rank of its metric differential then yields an exhaustive trichotomy:

• Rank at least two: dimension forces a physical tangent with positive Bell second derivative, contradicting local maximality.
• Rank one: projective-fiber rigidity prevents the strictly positive multiplier required by a strict separator.
• Rank zero: the behavior has an explicit bounded-transportation decomposition into deterministic local, hence qubit-PVM, behaviors.

No numerical Bell optimization, SDP bound, or finite POVM sampling is used in the closure theorem.

3. Permanent DOI records

The publication and its verification materials are preserved as five public Zenodo records. The standard publication PDF is the primary citation object.

4. Exact verification and reproducibility

The software archive contains the paper source and rendered PDFs, exact Bell coefficients and strategies, two independent symbolic verifiers, an exact rank-zero simulator, proof and dependency audits, a theorem-to-artifact map, and a one-command offline runner.

• Software archive — Python 3.14.6, SymPy 1.14.0, and Tectonic 0.16.9 reference environment.
• \(3\times2\) verifier — exact effects, Bell values, dual certificates, robust inequalities, and strict gap.
• \(2\times2\) closure verifier — 39 exact algebraic checks for the Lorentz metric, null-ray map, exceptional fibers, second variation, and bounded flow.
• Rank-zero simulator — exact deterministic decomposition on the symmetric trine instance.
• Website file hashes — checksums for the publication and line-numbered PDFs served here.

The executable checks verify finite exact identities and explicit constructions. They do not replace the manuscript arguments establishing compactness, convexification, extremal reduction, physical reconstruction, multiplier signs, or exhaustion of the rank cases.

5. Scope and limitations

• The equality is between shared-randomness convex hulls, not the raw fixed-dimensional strategy images.
• No same-state or operator-level simulation of arbitrary POVMs by PVMs is claimed.
• The \(3\times2\) construction proves a strict separation using lower and upper certificates; it does not determine either global optimum exactly.
• The theorem is specific to local dimension at most two and to linear Bell-function optimization.
• The paper was publicly archived before independent specialist peer review.

AI-assisted research and verification

OpenAI language models were used extensively in exploration, proof development, software generation, verification design, literature organization, and drafting. Alec Kriebel directed the work, selected the released claims, and is responsible for the decision to publish these materials. Model output was not treated as evidence by itself; the complete proofs and exact artifacts are supplied for independent scrutiny.

Suggested citation

Alec Kriebel, “Minimum Bell-Setting Complexity for Qubit POVM–PVM Separation,” version 1.1.0, Zenodo, 30 July 2026. doi:10.5281/zenodo.21699161.