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Yu–Tan conjectures on exact common information and exact channel synthesis

Let (X,Y)(X,Y) be jointly Gaussian with X,Y∼N(0,1)X,Y\sim\mathcal{N}(0,1) and correlation coefficient ρ∈(0,1)\rho\in(0,1). The exact common information of (X,Y)(X,Y) is conjectured to satisfy Cexact(X;Y)=12log⁡1+ρ1−ρ+ρ1+ρC_{\mathrm{exact}}(X;Y)=\frac{1}{2}\log\frac{1+\rho}{1-\rho}+\frac{\rho}{1+\rho}, where Cexact(X;Y)C_{\mathrm{exact}}(X;Y) denotes the minimum common-randomness rate required for exact synthesis of the joint distribution of (X,Y)(X,Y).

References

Primary source

arXiv

Additional references

information-theoryoptimal-transportprobability
Posted

Progress summary

Refreshed
Claimed solved

A new preprint claims to settle both conjectures about exact information sharing, but the result has not been independently verified.

Yu and Tan posed conjectures in 2020 concerning exact common information and the communication/shared-randomness tradeoff for exact channel synthesis. A newly listed preprint claims to establish both conjectures, including the formula and the complete admissible rate region.

Known results

  • Exact channel synthesis has multiletter characterizations and single-letter inner and outer bounds; these coincide for doubly symmetric binary sources (Yu and Tan, 2018).
  • For doubly symmetric binary sources, exact synthesis requires more communication than total-variation-approximate synthesis (Yu and Tan, 2018).
  • Exact common information is strictly larger than Wyner’s common information in general, with a characterization for doubly symmetric binary sources (Yu and Tan, 2018).
  • For finite alphabets, finite shared randomness can attain communication rate I(X;Y)I(X;Y); for jointly Gaussian sources, finite shared randomness cannot do so (Yu and Tan, 2018).

August 2026 claimed resolution

The preprint Exact Common Information and Exact Channel Synthesis for Correlated Gaussian Sources claims that both Yu–Tan conjectures are established, including the exact formula and admissible shared-randomness/communication-rate region. This is a claimed complete resolution, but the source is an unrefereed preprint and no independent verification or published confirmation was found.

Current status (as of August 2026): Both conjectures are claimed solved by an unrefereed preprint, while independent verification remains absent.

Sources

Solutions 0

No solutions have been posted yet.