Unique Information and Secret Key Decompositions

The unique information (UI) is an information measure that quantifies a deviation from the Blackwell order. We have recently shown that this quantity is an upper bound on the one-way secret key rate. In this paper, we prove a triangle inequality for the UI, which implies that the UI is never greater than one of the best known upper bounds on the two-way secret key rate. We conjecture that the UI lower bounds the two-way rate and discuss implications of the conjecture.