  We explore the structure of holographic entropy relations (associated with
'information quantities' given by a linear combination of entanglement
entropies of spatial sub-partitions of a CFT state with geometric bulk dual).
Such entropy relations can be recast in multiple ways, some of which have
significant advantages. Motivated by the already-noted simplification of
entropy relations when recast in terms of multipartite information, we explore
additional simplifications when recast in a new basis, which we dub the
K-basis, constructed from perfect tensor structures. For the fundamental
information quantities such a recasting is surprisingly compact, in part due to
the interesting fact that entropy vectors associated to perfect tensors are in
fact extreme rays in the holographic entropy cone (as well as the full quantum
entropy cone). More importantly, we prove that all holographic entropy
inequalities have positive coefficients when expressed in the K-basis,
underlying the key advantage over the entropy basis or the multipartite
information basis.
