Re-Deriving Quantum Entanglement from Classical Radon–Nikodým Hydrodynamics: A Formal Falsification Analysis of the Hydrodynamic Stability Hypothesis

In the paper's words

The Hydrodynamic Stability Hypothesis (HSH) conjectures that standard quantum mechanics, including multi-particle entangled states such as the Greenberger–Horne–Zeilinger (GHZ) state, can be axiomatically re-grounded in classical measure theory: wavefunctions are interpreted as Radon–Nikodým densities that relax into basins of attraction of classical probability measures under a local hydrodynamic transport, restoring local realism without superdeterminism. We subject this conjecture to a formal test. We state the hypothesis as four explicit axioms (classical probability space, local state variables, local continuous transport, free measurement settings) and attempt an explicit derivation of the GHZ state and its Bell-type correlations purely from those axioms.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The entry for arXiv:1110.0157v1 [3] states that 75 years after the term "entanglement" was coined, the connection between quantum and classical mechanics remains an open problem, and discusses, drawing on results in semiclassical systems, the fate of entanglement in a closed system as Planck's constant becomes vanishingly small; the summary is truncated at this point.

   from 2. Background and Related Work

2. A CHSH cross-check gives $2\sqrt{2} \approx 2.828427$ against the classical bound $2$.

   from Abstract

3. 122 140504] showed a continuous trade-off between quantum speedup and circuit depth — can be naturally and succinctly derived within a classical–quantum framework using quantum singular value transformations; the summary truncates before further details.

   from 2. Background and Related Work

4. The violation factor over the classical bound is $$\frac{2\sqrt{2}}{2} = \sqrt{2} \approx 1.414214.$$ Thus any local classical Radon–Nikodým model reproducing two-party entangled statistics is confined to $|S| \le 2$, while quantum mechanics reaches $2\sqrt{2} \approx 2.828427$; the gap is a factor of $\sqrt{2} \approx 1.414214$, confirming the three-party result at the two-party level.

   from 4.4 CHSH cross-check

5. Mermin bound and violation (Section 4.3): classical $|\langle P_1\rangle - \langle P_2\rangle - \langle P_3\rangle - \langle P_4\rangle| \le 2$; quantum value $M_{\mathrm{Q}} = 4$; gap $\Delta M = 2$; violation factor $4/2 = 2$.

   from 5. Results

6. CHSH cross-check (Section 4.4): classical bound $2$; quantum maximum $2\sqrt{2} \approx 2.828427$; violation factor $\sqrt{2} \approx 1.414214$.

   from 5. Results

How it is done

1. We show with full arithmetic that the Mermin GHZ argument forces any model satisfying the axioms to obey a pointwise product constraint $P_1 P_2 P_3 P_4 = +1$, while the quantum predictions require the corresponding expectation $-1$; the Mermin expression reaches the quantum value $M_{\mathrm{Q}} = 4$ against the classical bound $M_{\mathrm{LHV}} = 2$, a violation factor of $2$.

   from Abstract

2. The Radon–Nikodým theorem allows a measure absolutely continuous with respect to a base measure to be represented by a density; HSH proposes that the quantum wavefunction is, at the foundational level, such a density over a classical configuration space, and that preparation and measurement are stability phenomena: trajectories fall into attracting basins of these measures.

   from 1. Introduction

3. The entry for arXiv:2103.14759v3 [2] states that a quantum internet aims at distributing bipartite entanglement between distant nodes, but that multipartite entanglement between nodes may empower the quantum internet for additional or better applications in communications, sensing, and computation, and presents an algorithm for generating multipartite entanglement between distributed nodes.

   from 2. Background and Related Work

4. Given A1–A4, we ask whether the model can reproduce the quantum joint statistics of the GHZ state $$|\mathrm{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr)$$ under local Pauli measurements $X$ and $Y$ on each of the three qubits.

   from 3.2 Falsification protocol

What would falsify it

1. Any mismatch falsifies the strong form, because the hypothesis claims to re-derive the quantum statistics from the classical axioms.

   from 3.2 Falsification protocol

2. Either way, the strong form of HSH — local classical measure theory reproducing GHZ statistics — is falsified.

   from 4.5 The fork and the falsification verdict

3. The strong form of the hypothesis is therefore falsified: reproducing GHZ statistics forces either non-local variables or contradiction.

   from Abstract

Where the claim stops

1. What the falsification does and does not show.

   from 6. Discussion

2. It does not falsify weaker versions.

   from 6. Discussion

3. We flag this as the hypothesis's most likely retreat position, and it is exactly the position that mainstream non-local interpretations occupy; the value of the derivation is that it shows HSH cannot simultaneously keep locality, measurement freedom, and the GHZ correlations.

   from 6. Discussion

Where to read

1. Abstract217 words · 1 min
2. 1. Introduction509 words · 2 min
3. 2. Background and Related Work826 words · 4 min
4. 3. Methods371 words · 2 min
5. 4. Analysis1,178 words · 5 min
6. 5. Results223 words · 1 min
7. 6. Discussion598 words · 3 min
8. 7. Conclusion240 words · 1 min
9. Appendix A. Divergence report141 words · 1 min
10. Appendix B. Claim attribution178 words · 1 min