We prove subadditivity for the Nichols--Wootters family of entropies interpolating between the von Neumann entropy and the subentropy, with all spectra evaluated in one common ambient dimension. Beyond the von Neumann entropy, equality holds exactly when at least one marginal is pure. This settles the subadditivity conjecture concerning subentropy. Every member except the von Neumann entropy fails strong subadditivity, already on a maximally entangled two-qubit state tensored with a mixed qubit. Its strong-subadditivity defect takes both signs on quantum Markov states. For a pure bipartite state tensored with an ancillary state, we identify the defect with a separately concave product gap and determine its exact maximum under rank constraints. For arbitrary bipartite states, we obtain an explicit lower bound in the marginal purities. For two-qubit states, we determine the optimal coefficient multiplying the product of the smaller marginal eigenvalues in every common ambient dimension, including the subentropy coefficient $1/3$. A positive representation by Haar averages of compressed matrix entropies gives the pinching inequality underlying these results and also yields entropy-gain bounds for unital quantum channels.As an application, each entropy equals the classical communication capacity retained after a specified isotropic partial projective measurement, at a prescribed average input state, even when entangled block encodings are allowed.

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