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 $\sqrt{2}$ 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
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.
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
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.
The principle of Information Causality (IC) has served as a fundamental postulate in quantum information theory, establishing critical bounds on the strength of quantum correlations and differentiating the quantum formalism from generalized non-signaling theories. While IC successfully recovers the Tsirelson bound for the first Fourier harmonic of the bipartite correlation function, its ability to enforce the full positive semi-definiteness (PSD) of the correlation kernel for all higher-order harmonics has remained a persistent axiomatic gap. Traditional Random Access Code (RAC) protocols, reliant on fixed-basis encodings, are intrinsically blind to high-frequency negative harmonics. This paper provides an exhaustive analysis of the Eigenvector Protocol—a novel -bit RAC utilizing a probabilistic mixing strategy over the eigenvector components of the correlation kernel matrix—which definitively proves that IC enforces kernel positivity for all harmonics. Furthermore, this report extensively investigates the physical implementation of the Eigenvector Protocol utilizing state-of-the-art integrated photonics. We analyze the severe scalability limitations of bulk quantum optics and traditional Mach-Zehnder Interferometer (MZI) meshes, including phase noise, temporal distributed crosstalk, and quantization errors. Also available on Academia.edu: https://www.academia.edu/attachments/132330461/download_file?s=portfolio
Süß (2026) presents a concise yet technically ambitious five-page preprint that claims to close a longstanding axiomatic gap in the information-theoretic reconstruction of bipartite quantum correlations. By introducing an explicitly constructed N-bit Random Access Code (RAC) protocol—the "Eigenvector Protocol"—the author proves analytically and verifies numerically that the principle of Information Causality (IC) alone enforces the positive semi-definiteness (PSD) of all Fourier-harmonic components of the bipartite correlation kernel for every integer k ≥ 1. This result completes the derivation chain No-Signaling + Rotational Covariance + IC ⇒ E(Δθ) = −cos(Δθ) that had previously relied on partial or numerical evidence. The work is self-contained, device-independent in spirit, and directly addresses the open Mamun, SM (2026)……………………….1 Reviewing Süß (2026), "Information Causality Implies Kernel Positivity: A Complete Proof via the Eigenvector Protocol" question left by Pawłowski et al. (2009) and subsequent numerical scans (Süß, 2025, 2026a, 2026b). This paper outlines a detailed summary, methodological critique, comparison with prior literature, assessment of strengths and limitations, and discussion of broader implications. Mamun, SM (2026)……………………….2 Reviewing Süß (2026), "Information Causality Implies Kernel Positivity: A Complete Proof via the Eigenvector Protocol" Also available on Academia.edu: https://www.academia.edu/attachments/132327893/download_file?s=portfolio
Hilbert spaces, introduced in the early twentieth century as infinite-dimensional generalisations of Euclidean space, provide the rigorous mathematical framework for quantum mechanics and numerous applied disciplines. This paper offers a comprehensive examination of their historical development, axiomatic structure, and multifaceted applications, with particular emphasis on their foundational role in quantum mechanics. We then connect these structures to quantum information geometry, where the (projective) Hilbert space is endowed with natural Riemannian metrics—most notably the Fubini–Study metric for pure states and the Bures metric for mixed states—that quantify statistical distinguishability, parameter estimation precision, and geometric phases. By exploring classical information geometry as a precursor and highlighting quantum extensions, we illustrate how Hilbert-space geometry reveals deep insights into quantum phase transitions, metrology, and quantum technologies. Nuances such as separability, rigged Hilbert spaces for continuous spectra, and infinite-dimensional subtleties are addressed, alongside implications for machine learning via reproducing kernel Hilbert spaces (RKHS) and beyond. The analysis underscores the enduring interplay between pure mathematics and quantum physics. Mamun, SM (2026) …………………………………..1 Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132277960/download_file?s=portfolio