Structural Impossibility of Antichain-Lattice Partial Information Decomposition
This work identifies a fundamental representational limitation in PID for information theory researchers, showing that axiomatic fixes cannot resolve the incompatibility.
The paper proves that antichain-lattice Partial Information Decomposition (PID) is structurally impossible for three or more sources, as two systems can share identical antichain-indexed atoms yet have different mutual information. It introduces System Information Decomposition (SID) as a target-free alternative.
Partial Information Decomposition (PID) represents multivariate mutual information via antichain-lattice that aims to specify which source groups can recover which informational components of a target. For three or more sources, widely desired PID axioms become mutually incompatible. This is often treated as an axiomatic tuning issue. This paper argues that the obstruction is representational, rooted in the antichain indexing itself, so that purely axiomatic adjustments within an antichain-lattice structure cannot resolve it in general. We first introduce System Information Decomposition (SID) for the special target-free three-variable setting, obtaining a self-consistent entropy decomposition with an operational redundancy definition. More fundamentally, we then show that for general multivariate PID, there is no universal rule that recovers the decomposed mutual information from the antichain-indexed information atoms. In particular, two systems can share identical atoms regardless of any axioms while having different mutual information. These results reveal the limits of antichain-lattice and motivate relation-based foundations for multivariate information measures.