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Information Causality Enforces Kernel Positivity: A Complete Proof via the Eigenvector Protocol (Part 4)

Daniel Süß

Published Mar 30, 2026• 1 citation
Publisher

Abstract

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.

DOI
10.5281/zenodo.19332722

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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.
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
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
This work presents a complete, non-circular, and principle-based derivation of the quantum correlation law E(Δθ)=−cos⁡(Δθ)E(\Delta\theta) = -\cos(\Delta\theta)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)K(\theta, \phi) = -E(\theta - \phi)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)SO(2,1)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.
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 generalize the Eigenvector Protocol to the full Bloch sphere S². Using a dimensionally corrected IC-sum and multiple SO(3)-symmetric measurement ensembles, we prove that Information Causality (IC) rigorously enforces kernel positivity (aₗ ≥ 0) for all Legendre coefficients of the correlation function. For even l ≥ 2, a strictly stronger analytical bound holds: the maximum CHSH value satisfies Mₗ ≤ 2√2 − 2(√2−1)cₗ,₀ < 2√2, where cₗ,₀ > 0 is the constant Chebyshev term of Pₗ. For odd l ≥ 3, higher multipoles are strongly suppressed by rotational geometry and numerically confirmed to satisfy Mₗ < 2√2. A residual space for weakly positive Almost Quantum correlations remains, identifying a structural gap whose closure requires an additional extremality principle — developed in a companion paper. This paper extends the original Eigenvector Protocol from the circle S¹ to the full Bloch sphere S².
Information causality is a physical principle that constrains bipartite post-quantum nonlocal correlations from information-theoretic assumptions. A direct characterization of the correlations satisfying it, however, remains difficult because the defining entropic condition must hold for all admissible protocols in the corresponding task. Here, we formulate the single-copy information causality problem as an optimization over all protocol parameters and simplify it by proving a deterministic reduction. We then show that generic adaptive multi-copy protocols can be represented as single-copy protocols applied to an effective behavior obtained by local wirings. As a consequence, for families of correlations closed under wirings, excluding single-copy violations suffices to exclude all finite-copy adaptive-protocol violations. We formulate the resulting membership and set-inclusion problems as optimization problems and discuss possible semidefinite programming relaxations. As an application, we study expansions of the information causality functional around zero-capacity channels, generalizing a method recently employed by Gachechiladze and Miklin to derive the Tsirelson--Landau--Masanes boundary from information causality. For expansions centered on channels whose output distribution has full support, imposing the second-order constraints for all single-copy protocols characterizes exactly the set of macroscopic locality in the CHSH scenario. Expanding instead around appropriate constant-output channels with perturbations that change the channel support, we show that every single-copy IC violation can be witnessed already at first order with a channel of arbitrarily small capacity.