Entropy subadditivity

In plain terms

Here is the problem. Entropy measures missing information, and for quantum systems the basic sanity check is subadditivity: the entropy of a joint system is at most the sum of the entropies of its parts. Correlations can only make the whole MORE predictable than its parts suggest, never less.

Every textbook states it; the proof leans on two steps that are easy to say and laborious to formalise. Klein's inequality -- relative entropy is nonnegative -- needs a careful spectral argument with a doubly-stochastic overlap matrix. And the logarithm of a tensor product splitting into a sum needs the operator functional calculus to commute with the Kronecker structure, which is exactly the step informal proofs wave through.

Both are done, so the inequality stands on checked ground, including for pure entangled states -- the case where the whole has zero entropy and the parts do not, which is the quantum signature classical entropy forbids.

In CSD

The entropy tier serves the programme's thermodynamics track and its decoherence accounting: entropy production in measurement, the entropy ledger of the chaos diagnostics, and the Landauer touchpoint all cite the von Neumann entropy machinery this inequality anchors.

Deferred pieces are named rather than hidden: the strong version -- subadditivity with three parties -- is proved conditionally on the data processing inequality, stated as an explicit hypothesis, and the triangle inequality's purification construction is recorded as the missing step.

Mathematically

vonNeumannEntropy_subadditive: S(rho_AB) <= S(rho_A) + S(rho_B) for a bipartite density with positive-definite marginals, the joint state merely PSD. Route: Klein's inequality via the doubly-stochastic overlap expansion; the Kronecker-log split log(A (x) B) = log A (x) I + I (x) log B via a diagonalisation-respecting functional calculus; and the reduced-trace identities collapsing the cross terms. Araki-Lieb's one side is proved under its stated hypotheses.