This work presents a complete, non-circular, and principle-based derivation of the quantum correlation law $E(\Delta\theta) = -\cos(\Delta\theta)$ from three operational postulates: no-signaling with fair marginals, rotational covariance, and Information Causality (IC). The central contribution is the proof that IC alone enforces the kernel positivity (positive semidefiniteness) of the correlation matrix $K(\theta, \phi) = -E(\theta - \phi)$. This closes a long-standing axiomatic gap, as prior derivations often required kernel positivity as an independent mathematical assumption.Combined with the Herglotz theorem and Tsirelson saturation, this yields the unique harmonic form of the cosine law without additional ad-hoc postulates. Furthermore, numerical analysis of the hyperbolic extension ($SO(2,1)$) reveals that while IC still implies kernel positivity in non-Euclidean geometries, the Tsirelson bound can be exceeded. This suggests a deep interplay between spacetime geometry and the observed limits of quantum correlations.
This work presents a complete, non-circular, and principle-based derivation of the quantum correlation law E(Δθ)=−cos(Δθ) from three operational postulates: no-signaling with fair marginals, rotational covariance, and Information Causality (IC). The central contribution is the proof that IC alone enforces the kernel positivity (positive semidefiniteness) of the correlation matrix K(θ,ϕ)=−E(θ−ϕ). This closes a long-standing axiomatic gap, as prior derivations often required kernel positivity as an independent mathematical assumption.Combined with the Herglotz theorem and Tsirelson saturation, this yields the unique harmonic form of the cosine law without additional ad-hoc postulates. Furthermore, numerical analysis of the hyperbolic extension (SO(2,1)) reveals that while IC still implies kernel positivity in non-Euclidean geometries, the Tsirelson bound can be exceeded. This suggests a deep interplay between spacetime geometry and the observed limits of quantum correlations.
This review critically examines Daniel Süß's March 2026 preprint, which advances a principle-based reconstruction of quantum bipartite binary correlations. By proving that the Information Causality (IC) principle alone implies kernel positivity (positive semidefiniteness of the correlation matrix), Süß eliminates a long-standing ad-hoc assumption in prior axiomatic derivations. Combined with no-signaling, rotational covariance, and the Herglotz theorem, this yields the unique quantum correlation function E(Δθ) = −cos(Δθ) purely from operational postulates. The work further derives the maximum-entropy joint probability distribution and extends the framework to hyperbolic geometry SO(2,1), revealing that while IC still enforces positivity, the Tsirelson bound can be exceeded. This review synthesises the paper's postulates, proofs, and results; provides algebraic and geometric interpretations; surveys related literature in information-theoretic reconstructions of quantum mechanics; and discusses implications, limitations, experimental prospects, and future directions. The analysis affirms the novelty and rigor of the derivation while highlighting its deeper connection between information theory, correlation structure, and spacetime geometry. Mamun, SM (2026)……………………………..1 A Comprehensive Review of "Information Causality as the Sole Origin of the Quantum Correlation Law" by Daniel Süß (Vol-I) Closing the Axiomatic Gap in Operational Derivations of Quantum Correlations Also available on Academia.edu: https://www.academia.edu/attachments/132230261/download_file?s=portfolio
The sustained axiomatic endeavor to derive quantum mechanics from minimal operational principles has been fundamentally limited by the need to ad-hoc postulate kernel positivity (Gram representability). This comprehensive work details the closure of this logical gap, proving that Information Causality (IC), when combined with no-signaling, rigorously and natively implies kernel positivity, thereby completing the operational derivation of the unique quantum cosine correlation law. Moving beyond bipartite, Euclidean regimes, the report systematically explores the tertiary implications of these informational axioms across three frontiers: (1) Multipartite Information Causality, which imposes elemental entropic inequalities (strong subadditivity, weak monotonicity) within coexisting sets of complex communication networks, rigorously enforcing the monogamy of nonlocality and enhancing the security of Device-Independent Quantum Key Distribution (DI-QKD). (2) Geometric Selectors, demonstrating that the 2 Tsirelson bound is a geometric signature of flat Euclidean space, as non-Euclidean hyperbolic spaces permit Information Causality to support supra-quantum correlations. This geometric constraint is validated by experimental tabletop realizations of the AdS/CFT correspondence in macroscopic hyperbolic electrical circuits. Also available on Academia.edu: https://www.academia.edu/attachments/133325508/download_file?s=portfolio
Key Takeaways ● ● ● ● ● Completion of Quantum Axiomatics: The long-standing axiomatic gap in quantum mechanics is closed by proving that Information Causality (IC), combined with no-signaling, rigorously mandates kernel positivity (positive semidefiniteness), thereby deriving the unique quantum correlation law (cos(θ)) without requiring ad-hoc Hilbert space assumptions. Geometric Nature of Quantum Limits: The Tsirelson bound (2) is not an absolute information-theoretic limit, but rather a geometric signature of flat Euclidean space. In non-Euclidean hyperbolic spaces (H2), Information Causality permits correlations that significantly exceed the quantum limit (supra-quantum violations). Holographic Reality is Experimentally Accessible: The relationship between geometric curvature and information is physically confirmed by demonstrating the holographic scaling of entanglement entropy (via the Ryu-Takayanagi formula) in macroscopic, tabletop hyperbolic electrical circuit lattices that simulate Anti-de Sitter (AdS) and BTZ black hole spacetimes. Unified Gravity Theory (CETΩ): The Causal-Informational Completion of Gravity (CETΩ) is a strictly 4D framework that replaces classical spacetime substance with a dynamic, emergent network of causal links. The theory elegantly regularizes black hole singularities and guarantees quantum unitarity across all scales via the spectral positivity of its nonlocal retarded kernel. Resolution of Cosmological Problems: The CETΩ framework introduces the "texon" scalar field, which unifies the dark sector (acting as both dark matter and dark energy) and provides a precise, late-time informational modification that alleviates the problematic Hubble tension.
Part 6 of this record supersedes Parts 1-5. Earlier parts conjectured that Information Causality (IC) alone enforces the positive-semidefinite (PSD) structure of translation-invariant correlation kernels; the Sole-Origin and Eigenvector versions (Parts 1, 4, 5) claimed constructive proofs. This paper withdraws those proofs and replaces them with what can be proven. Two structure theorems show the withdrawal is structural, not incidental. The Barrier Lemma: any protocol whose measurement settings are independent of the encoded data satisfies S ≤ E(0)²‖u‖₂² ≤ 1, for every Bob strategy and every resource model (verified by exhaustive maximization over all Bob functions at N=4). The Pyramid Factorization Theorem: the Pawłowski concatenation satisfies S_{2ⁿ} = Π levels ≤ (sup S₂)ⁿ, so nesting can never rescue a kernel that a single level cannot detect. Two positive results take their place. A visibility dichotomy: at sub-unit visibility, explicitly non-PSD kernels are quantum-realizable via a Tsirelson factorization with sign-flipped harmonics, so no correlation principle can enforce PSD there; at unit visibility (E(0) = −1), Cauchy-Schwarz rigidity forces single-family Gram structure. The main theorem: Macroscopic Locality implies kernel positivity at unit visibility (via the Q¹ characterization of ML plus the rigidity argument). The axiomatic gap open since Pawłowski et al. (2009) is thereby closed, by Macroscopic Locality. Corollary: every unit-visibility non-PSD kernel lies outside the Almost-Quantum set, confirmed numerically by robust NPA-1+AB semidefinite programming (N = 7 settings, Clarabel, PSD control at 10⁻⁹). Whether the strictly weaker principle IC suffices remains open, and is sharpened by seven benchmark kernels, unit visibility, non-PSD, provably outside Almost Quantum, that resist every IC protocol tested: optimized N=2/3 functionals, general random-access codes up to four bits, the full concatenation pyramid with free angles, linear parity encodings, entropic and mutual-information functionals, and continuous-angle protocols. Either a fundamentally new protocol detects them (IC ⇒ PSD), or one of them satisfies IC (IC strictly weaker than ML here). Both outcomes are discoveries; the benchmark coefficients are given in full and machine-readable. An open verification suite (6 self-contained Python scripts, MIT license, ~90 s runtime) is included and reproduces every numerical claim: kernel validity, the Barrier Lemma (three exact checks), the Pyramid Factorization, the Tsirelson factorization of the dichotomy, the NPA-1+AB distances for all seven benchmarks, and the H² bound. Following the documentation practice of this series, all withdrawn claims and their precise failure modes are recorded, including one hypothesis raised and refuted during review of this work.
Information causality (IC) posits that communicating m classical bits cannot give the receiver access to more than m bits of information about the sender's independent data. Although IC holds in quantum theory and excludes a broad class of supra-quantum correlations, its validity for the almost-quantum set has remained open. This semidefinite relaxation of the quantum set satisfies nontrivial communication complexity, no advantage for non-local computation, macroscopic locality, local orthogonality, and numerical evidence has suggested that it may also satisfy IC. Here we show that almost-quantum correlations violate IC, and do so without bound. Unlike earlier violations of IC, which rely on biased guesses, ours is driven by zero-error certificates: each is rare but never wrong, carries information linear in its probability, and vouches for many of the sender's bits at once. This establishes a sharp dichotomy: either IC is a valid physical principle that excludes these correlations, or any nonsignalling theory realizing them admits no mutual information obeying consistency, data processing and the chain rule. In the latter case, we discuss why the chain rule is the natural property to relinquish, and how this possibility relates to histories-based formulations and to locality defined relative to the state rather than presupposed in a fixed spacetime.
We prove that quantum mechanics is the unique optimal physical theory under Information Causality (IC): it is the only rotationally covariant bipartite correlation theory that both respects IC and maximises non-local coordination. Concretely, among all IC-admissible positive semi-definite (PSD) kernels E(d) = −∑ₗ≥₁ aₗPₗ(cos d) with aₗ ≥ 0, the quantum correlation E(d) = −cos d is the unique maximiser of the CHSH value. The proof combines the Eigenvector Protocol (IC implies aₗ ≥ 0) with the Tsirelson uniqueness result into a single optimality statement. No second independent axiom is required: Tsirelson saturation is not an additional assumption but a consequence of optimality under IC. This provides the tightest known single-principle characterisation of quantum correlations, establishing that quantum mechanics is not merely consistent with IC — it is the optimal realisation of IC.