Alec Kriebel

Minimum Bell-Setting Complexity for Qubit POVM–PVM Separation

By Alec Kriebel, independent researcher.
Published · Archive label v1.1.0 · Unrefereed preprint

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Plain-English explanation · Read on this page · Collaboration

Abstract

Vértesi and Bene exhibited a 3 × 2 Bell scenario in which qubit positive-operator-valued measurements (POVMs) outperform qubit projection-valued measurements (PVMs). We prove the matching setting lower bound. For arbitrary finite, input-dependent output alphabets, the shared-randomness convex hulls of bipartite qubit-POVM and qubit-PVM behaviors are equal when both parties have two inputs. Equivalently, no two-input linear Bell functional separates the two convexified sets.

We also give a rational 3 × 2 functional, an exact qubit-POVM strategy, and a global analytic qubit-PVM upper bound with a strict exact gap. Hence 3 × 2 is the minimum input architecture, up to exchanging the parties.

The equality proof combines a one-binary-party Lorentz-cone simulation, an extremal binary–ternary reduction, a physically exact Lorentz incidence manifold, finite-POVM duality, and a rank trichotomy based on the incidence second variation. The result concerns convexified behavior sets; it does not assert equality of the raw nonconvex strategy images or same-state simulation.

The idea in plain English

In a Bell experiment, two separated observers each choose a measurement and record an outcome. Generalized quantum measurements, called POVMs, allow more possibilities than projective measurements, called PVMs. This paper asks how many measurement choices are needed before that extra freedom changes the observable correlations when each observer’s quantum system is a qubit.

With only two choices for each observer, the paper shows that any probability table produced by qubit POVMs can also be produced by a shared-randomness mixture of qubit PVM strategies. This holds for every finite number of declared outcomes, even when the outcome lists differ between measurements.

The mixture may use different states and measurements on different runs. The result concerns reproducing the joint probabilities; it does not turn every individual POVM into a projective measurement on the original state.

Three choices on one side and two on the other are enough for a strict advantage. Vértesi and Bene established that phenomenon in 2010. The proposed advance here is the matching two-choice lower bound, together with an explicit rational Bell expression and exact bounds certifying a three-by-two gap.

The precise result

For every bipartite architecture with two inputs per party and arbitrary finite, input-dependent output alphabets (A₀, A₁; B₀, B₁), conv Q₂ᴾᴼⱽᴹ(A, B) = conv Q₂ᴾⱽᴹ(A, B). Each strategy has local Hilbert-space dimension at most two. The convex hull permits a shared classical variable to select the state and all measurements simultaneously.

The PVM class includes zero projectors, deterministic measurements, all labelled support patterns, and stochastic output postprocessing. Dimension-increasing ancillary systems are excluded. Equivalently, every linear Bell functional with two inputs per party has the same qubit-POVM and qubit-PVM optimum.

For the rational three-by-two Bell functional in equation (1), the paper gives an attained POVM value L₀ = 20√2 + 16/25 and a global PVM upper bound U = 20√2 + 3/5 + (4 + 3√2)/250. The certified gap is L₀ − U = 3(2 − √2)/250 > 0.

The witness has output architecture (2, 2, 3) for one party and (2, 2) for the other. Together with the two-input equality and the locality of one-input scenarios, it gives the minimum input architecture 3 × 2, up to exchanging the parties. The numbers 3 and 2 count measurement settings, not outcomes.

Why the result holds

If one observer has only binary outcomes, the other observer’s steered states can be compressed into a three-dimensional cone. Decomposing relations in that cone gives one common mixture of complete projective strategies, preserving the whole probability table.

A hypothetical two-input Bell separator can then be reduced to one binary and one ternary measurement on each side. The remaining physical strategies admit exact Lorentz-metric coordinates. At a strict nonprojective maximum, duality forces positive multipliers, and an exhaustive rank analysis either produces an improving physical direction or an explicit projective simulation.

For the three-by-two witness, a CHSH term constrains the state and measurement geometry while an additional three-outcome measurement gains a state-discrimination advantage. Exact algebra bounds every projective alternative, including zero effects and deterministic cases.

Scope and limitations

The equality is between shared-randomness convex hulls. It does not establish equality of the raw nonconvex strategy images, a simulation using the same state, or projective simulability of an individual POVM as an operator-valued measurement.

The classification is bipartite and restricted to local Hilbert spaces of dimension at most two, finite output alphabets, and ordinary single-round Bell behaviors. It does not extend here to higher dimensions, multiple parties, or sequential measurements.

The three-by-two certificates provide an attained lower bound and a global upper bound. Neither is claimed to be the exact global optimum. The paper also derives no private-randomness entropy guarantee.

The code checks finite identities and explicit constructions. Compactness, extremal reduction, physical reconstruction, multiplier positivity, and exhaustion of the rank cases require the written proof. The paper remains unrefereed and was developed with extensive AI assistance.

Status and AI use

This is an unrefereed preprint developed with extensive generative-AI assistance in mathematical exploration, derivation, proof analysis, software, literature work, and writing. The human author accepts responsibility for the released claims. The supplied proofs and exact artifacts support independent scrutiny; they do not constitute independent expert peer review.

The manuscript is dated July 2026. The deposited PDF filename and companion archives use the label v1.1.0; the Zenodo record does not supply a numbered version.

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Supporting materials

Collaborate on this result

I welcome substantive collaboration on an independent review of the arbitrary-output two-input theorem, particularly the physical reconstruction and exceptional-rank arguments, or on preparing the result for journal submission. Expertise in Bell correlations, convex geometry, or quantum measurement theory would be especially useful.

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

Contact Alec

Cite this paper

Alec Kriebel. (2026). Minimum Bell-Setting Complexity for Qubit POVM–PVM Separation (Manuscript dated July 2026; archive label v1.1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.21699161

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Background and prior work