  Strong subadditivity goes beyond the tensored subsystem and commuting
operator models. As previously noted by Petz and later by Araki and Moriya, two
subalgebras of observables satisfy a generalized SSA-like inequality if they
form a commuting square. We explore the interpretation and consequences in
finite dimensions, connecting various entropic uncertainty relations for
mutually unbiased bases with the positivity of a generalized conditional mutual
information (CMI), and with inequalities on relative entropies of coherence and
asymmetry. We obtain a bipartite resource theory of operations under which the
two subalgebras are respectively invariant and covariant, with CMI as a
monotone, and generalized non-classical monotones based on squashed
entanglement and entanglement of formation. Free transformations support
conversion between entanglement and uncertainty-based configurations, as "EPR
<-> 2UCR." Our theory quantifies the common non-classicality in entanglement
and uncertainty, implying a strong conceptual link between these fundamentally
quantum phenomena.
