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arxiv: 2406.14153 · v1 · submitted 2024-06-20 · 🪐 quant-ph · math-ph· math.MP· math.PR

On random classical marginal problems with applications to quantum information theory

Ankit Kumar Jha , Ion Nechita This is my paper

Pith reviewed 2026-05-24 00:33 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPmath.PR
keywords classical marginal problemrandom distributionsCHSH scenarioBell-Wigner scenariovolume ratioslocal polytopenon-signaling polytopequantum correlations
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The pith

Random bivariate distributions on graph edges are consistent with a global joint distribution with probabilities estimated from polytope volumes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models the classical marginal problem using graphs where vertices carry fixed binary probability distributions and edges carry random bivariate distributions that match the vertex marginals as inputs. It derives estimates for the probability that these edge marginals can be realized by some joint distribution over all vertices. The motivation comes from Fine's theorem in quantum mechanics, and the work examines the CHSH and Bell-Wigner graphs in detail to obtain explicit ratios of the volumes of the associated local and non-signaling polytopes. A reader would care because these probabilities and ratios describe how often observed pairwise correlations admit a classical joint explanation.

Core claim

By encoding marginal constraints as graphs with fixed vertex distributions and random edge distributions, the authors estimate the probability that a consistent joint distribution on the graph exists and compute the volume ratios between the local and non-signaling polytopes for the CHSH and Bell-Wigner scenarios.

What carries the argument

Graph encoding of the marginal problem with fixed binary vertex distributions and random bivariate edge distributions matching those marginals.

Load-bearing premise

The model of sampling random bivariate distributions on edges with fixed vertex marginals is representative of typical instances of the classical marginal problem on the studied graphs.

What would settle it

A Monte Carlo sampling experiment over many random bivariate distributions on the CHSH graph that measures the actual fraction admitting a joint distribution and finds a value significantly different from the reported volume ratio.

Figures

Figures reproduced from arXiv: 2406.14153 by Ankit Kumar Jha, Ion Nechita.

Figure 1
Figure 1. Figure 1: The scaled local polytope slice L˜ t , for 0 < t ≤ 1/3 (left). In the right panel, we partition it into a triangular pyramid and a regular tetrahedron. Hence, the volume of the combined region is: ∀t ∈ (0, 1/3] vol(L˜ t) = 1 3 + 1 6 = 1 2 . (17) [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The scaled local polytope slice L˜ t , for 1/3 < t ≤ 1/2 (in red). The yellow region represents the difference between this case and the case when t ≤ 1/3, see [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Volume ratio between the local polytope slice and the non-signaling polytope slice, in the symmetric (pi = t) case. The curve is symmetric with respect to t = 1/2, and constant (= 1/2) for t ≤ 1/3. 6.2. Skewed slices (p1, p2, p3) = (t, t, 1/2 − t). We now consider a different, non-symmetric, slice through the local and non-signaling polytopes of the triangle graph. For a parameter t ∈ [0, 1/2], we consider… view at source ↗
Figure 4
Figure 4. Figure 4: The scaled local polytope slice L˜ t , for 0 < t ≤ 1/6 (left). In the right panel, we show the two triangular pyramids removed from the unit cube to obtain figure on the left. Both conv{(1, 0, 0),(0, 0, 1),(1, 1, 1),(1, 0, 1)} and conv{(1, 0, 0),(0, 0, 1),(1, 1, 1),(1, 1, 0)} are tri￾angular pyramids with volume 1/6 as can be seen in figure 4. The scaled slice L˜ t is just the unit cube with these two regi… view at source ↗
Figure 5
Figure 5. Figure 5: The scaled local polytope slice L˜ t , for 1/6 < t ≤ 1/4 (in red). The yellow region represents the difference between this case and the case when t ≤ 1/6, see [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The scaled local polytope slice L˜ t , for 1/4 < t ≤ 1/2 (left). In the right panel, we show the three triangular pyramid regions removed from the unit cube to get figure on the left. Hence, we have: ∀t ∈ (1/4, 1/2] vol(L˜ t) = 1 − (1/(2t) − 1) 2 (29) Proposition 6.3. The volume ratio of skewed slices (p1 = p2 = t, p3 = 1/2 − t) between the local and no-signaling polytopes corresponding to the triangle gra… view at source ↗
Figure 7
Figure 7. Figure 7: Volume ratio between the local polytope slice and the non-signaling polytope slice, in the skewed (p1 = p2 = t, p3 = 1/2 − t) case. p1 p2 p4 p3 q12 q23 q34 q14 The V -representation of L(K2,2) = COR(K2,2) is given by the convex hull of all rows of the truth table presented in [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The x23 = x34 slices of L(p, K2,2). Left: t ∈  0, 1 3 i . Right: t = 0.4; the yellow sections are the volumes removed due to inequalities (35). Thus, combining the results of this section, we obtain our main result. This is plotted against numerical result in [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: vol(L(p,K2,2)) vol(N(p,K2,2)) as a function of t In terms of the CHSH game, Proposition 7.1 can be interpreted as follows. Consider Alice and Bob sharing a randomly sampled no-signaling box with the condition that the marginal distribu￾tion of each of their own questions are fixed to be t. In such a case, the highest probability of the randomly sampled box being non-local occurs when t = 1/2. In fact, the … view at source ↗
Figure 10
Figure 10. Figure 10: The volume ratio vol Lt/ vol Nt plotted as a function of t for Cn. The legend shows the value of n for each plot. We see that as n increases, the volume ratio tends to 1. p1 p2 p4 p3 q12 q23 q34 q14 q13 q24 [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The K4 graph with qi = pi in the expansion. Now, since in our case, we do not deal with hypergraphs, we are limited to an intersection of maximum two events (recall that contexts are of maximum size 2). Thus, we modify the Inclusion-Exclusion inequality in our case as : pi1i2···ik = X 2 r=1 (−1)r+1 X j1,...,jr∈{i1,...,ik} qj1···jr ≥ 0. (44) We start be recalling a general result about complete graphs. Pro… view at source ↗
Figure 12
Figure 12. Figure 12: The tree structure generated on gluing graphs As an example, notice that K4 − e can be formed by gluing two K3 along an edge. Infact, the two sets of inequalities (51a, 51c, 52a, 52b) and (51b, 51d, 52c, 52d) are isomorphic to (12) over some vertex relabellings. Hence, H-representation of L(K4 − e) can be obtained from the H-representation of L(K3) [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: K4 −e formed by joining two K3 as shown in (A). The red components in (B) show the common induced subgraph. A direct consequence of (10.1) and (10.2) is that all the non-trivial inequalities (39) for L(Cn) can be obtained just by iteratively gluing together K3 along an edge and removing it as demonstrated in [AQB+13]. Lets start by studying, how Proposition 10.1 and Proposition 10.2 affect the volume rati… view at source ↗
Figure 15
Figure 15. Figure 15: Plots of volume ratio of symmetric slices given by, vol L(p = (t, t, t, t), G)/ vol N(p = (t, t, t, t), G) as a function of t for different graphs with 4 vertices. The graphs are shown on the lower right corner of each plot. Graph G tw(G) τ (G) ρ0+(G) ρ1/2 (G) K4 3 1 4 5 36 2 45 K4 − e 2 1 3 17 60 2 15 K2,2 2 1 3 5 6 2 3 pan 2 1 3 1 2 1 3 [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Plots of volume ratio of symmetric slices given by vol L(p = (t, t, t, t, t), G)/ vol N(p = (t, t, t, t, t), G) as a function of t for different graphs with 5 vertices. The graphs are displayed on the lower right corner of each plot. The plots for graphs K5 − e and K5 are excluded due to computational intractability [PITH_FULL_IMAGE:figures/full_fig_p042_16.png] view at source ↗
read the original abstract

In this paper, we study random instances of the classical marginal problem. We encode the problem in a graph, where the vertices have assigned fixed binary probability distributions, and edges have assigned random bivariate distributions having the incident vertex distributions as marginals. We provide estimates on the probability that a joint distribution on the graph exists, having the bivariate edge distributions as marginals. Our study is motivated by Fine's theorem in quantum mechanics. We study in great detail the graphs corresponding to CHSH and Bell-Wigner scenarios providing rations of volumes between the local and non-signaling polytopes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper encodes the classical marginal problem on graphs with fixed binary distributions on vertices and random bivariate distributions (with those marginals) on edges. It provides estimates on the probability that a joint distribution on the vertices exists that is consistent with the given edge marginals, motivated by Fine's theorem, and computes explicit ratios of the volumes of the local polytope to the non-signaling polytope for the graphs corresponding to the CHSH and Bell-Wigner scenarios.

Significance. If the sampling measure is the natural volume measure on the space of bivariate distributions with fixed marginals, the probability estimates and polytope-volume ratios would give quantitative information on the typicality of classical marginal consistency in Bell scenarios, which is of interest for quantum information applications.

major comments (2)
  1. [Methods / sampling section (near the description of random edge distributions)] The probability estimates and volume ratios in the abstract are only geometrically meaningful if the random bivariate distributions are sampled from the uniform (Lebesgue) measure on the simplex of distributions with fixed vertex marginals. The manuscript must explicitly state the sampling procedure (including any reparameterization, Dirichlet parameters, or rejection method) and demonstrate that it induces the uniform measure; otherwise the reported probabilities and ratios do not support the geometric claims.
  2. [Section describing the CHSH and Bell-Wigner graphs] For the CHSH and Bell-Wigner graphs, the encoding of the marginal constraints and the definition of the local vs. non-signaling polytopes must be shown to match the standard Bell polytope constructions; any deviation in the graph encoding would affect the validity of the volume ratios.
minor comments (1)
  1. [Abstract] Abstract contains the typo 'rations of volumes' (should be 'ratios of volumes').

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will incorporate clarifications in a revised version.

read point-by-point responses

  1. Referee: [Methods / sampling section (near the description of random edge distributions)] The probability estimates and volume ratios in the abstract are only geometrically meaningful if the random bivariate distributions are sampled from the uniform (Lebesgue) measure on the simplex of distributions with fixed vertex marginals. The manuscript must explicitly state the sampling procedure (including any reparameterization, Dirichlet parameters, or rejection method) and demonstrate that it induces the uniform measure; otherwise the reported probabilities and ratios do not support the geometric claims.

    Authors: We agree that the geometric claims require the sampling to induce the uniform Lebesgue measure on the simplex of bivariate distributions with fixed marginals. The original manuscript describes a random generation procedure for the edge distributions but does not explicitly verify equivalence to the uniform measure. In the revision we will add a dedicated paragraph in the methods section detailing the exact sampling algorithm (a reparameterization of the simplex coordinates) together with a short proof that the induced measure is Lebesgue, thereby supporting the reported probabilities and volume ratios. revision: yes

  2. Referee: [Section describing the CHSH and Bell-Wigner graphs] For the CHSH and Bell-Wigner graphs, the encoding of the marginal constraints and the definition of the local vs. non-signaling polytopes must be shown to match the standard Bell polytope constructions; any deviation in the graph encoding would affect the validity of the volume ratios.

    Authors: The graph encoding was constructed precisely to reproduce the marginal constraints of the standard CHSH and Bell-Wigner Bell polytopes via Fine's theorem. Nevertheless, to remove any ambiguity we will insert an explicit subsection that maps each graph vertex/edge constraint to the corresponding no-signaling and locality conditions in the usual Bell-polytope formulation, confirming that the local and non-signaling polytopes coincide with the standard ones and that the computed volume ratios are therefore valid. revision: yes

Circularity Check

0 steps flagged

No circularity; direct estimates and volume computations on explicit graphs

full rationale

The paper encodes marginal problems on graphs with fixed vertex distributions and random bivariate edge marginals, then derives existence probabilities and local/non-signaling polytope volume ratios for CHSH and Bell-Wigner graphs via explicit constructions and probabilistic estimates. No step reduces by the paper's own equations to a fitted parameter renamed as prediction, a self-definition, or a load-bearing self-citation chain; the central quantities are computed directly from the graph model without importing uniqueness theorems or ansatzes from prior author work. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard results from probability theory (existence of joints given marginals) and convex geometry (polytopes of local and non-signaling distributions). No free parameters, ad-hoc axioms, or new postulated entities are mentioned.

axioms (1)
  • standard math Standard theorems on the existence of joint distributions consistent with given marginals and on the geometry of marginal polytopes.
    Invoked when the graph encoding and volume ratios are defined.

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