A version-one arXiv preprint reports exact answers to two questions about correlated Gaussian sources. It concludes that the Yu–Tan conjectured formula for exact common information is correct and that the conjectured rate region for exact channel synthesis is also the full region for the model studied.
The work is a proof-based information-theory study, not an experiment. It analyzes exact synthesis codes and the standard bivariate Gaussian distribution πρ = N(0, Σρ), with 0 ≤ ρ < 1. There is no empirical participant sample or population behind the result; the evidence comes from mathematical probability models and codes.
An exact formula, including a positive-correlation difference
For every 0 ≤ ρ < 1, the paper gives the exact common information as CExact(πρ) = 1/2 ln((1+ρ)/(1−ρ)) + ρ/(1+ρ). In this setting, common information is the quantity associated with the shared randomness used to coordinate the two Gaussian variables. The expression is presented as an exact theorem, rather than as an estimate with statistical uncertainty.
For ρ > 0, the exactness penalty relative to Wyner’s common information is ρ/(1+ρ), and the exact common-information value is strictly larger. The paper therefore gives a precise mathematical comparison between the exact quantity and Wyner’s comparator in the positive-correlation case.
That qualification matters. The added difference is stated for positive correlation, while the exact formula covers the modeled range from zero up to, but not including, one. The paper also reports a scalar extension to negative correlation in which ρ is replaced by |ρ| within the stated range.
The result comes from matching bounds
The first conjecture is settled through a multiletter converse. In plain language, the argument considers exact synthesis across a block of uses and supplies a lower bound that every exact synthesis code must meet. Combined with Yu–Tan’s upper bound, that lower bound confirms the first conjecture.
The proof uses several analytical ingredients: an optimal-transport representation of worst-case Gaussian cross-entropy, Fathi’s Gaussian transport inequality, and a covariance-based determinant inequality. These are mathematical steps in the derivation, not measurements taken from an observed system.
The paper also reports a conditional-entropy bound of n log(2πe(1−ρ)) for every exact Gaussian code. This supporting bound is part of the proof-based characterization of the codes, rather than a result inferred from a statistical sample.
The same pattern appears in channel synthesis
The second conjecture concerns the rate region for exact channel synthesis: the combinations of shared-randomness and communication rates allowed by exact codes. The paper’s outer-bound result places the exact synthesis rate region inside the conjectured region over the stated correlation range, and its final theorem establishes equality between the two regions.
The paper separately compares this result with total-variation channel synthesis. For ρ > 0, exact channel synthesis requires strictly higher rates than total-variation synthesis. This is a rate comparison between two theoretical synthesis criteria.
A theorem for a defined model
The findings characterize the specified scalar Gaussian setting. They do not establish the same formulas for non-Gaussian source distributions, and they do not report finite-blocklength implementation benchmarks or empirical rate measurements. Readers should therefore treat the results as a mathematical characterization of exact codes and rates and regions, not as evidence about human participants or a real-world population.
The document is an arXiv version-one preprint dated 26 Aug 2026. The supplied metadata reports no journal or repository venue for the work.
Paper data and sources
Original title: Exact Common Information and Exact Channel Synthesis for Correlated Gaussian Sources
Authors: Lei Yu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text