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Write a bipartite pure state in fixed product bases as $|\psi ⟩={\sum }_{ij}{M}_{ij}|i⟩|j⟩$. The Schmidt decomposition theorem, equivalently the singular value decomposition of $M$, says that its Schmidt rank equals $\mathrm{rank}M$. A branch of local operations is a product operator $A\otimes B$, so its coefficient matrix isThe elementary rank inequality $\mathrm{rank}(AM{B}^{\mathsf{T}})\le \mathrm{rank}M$ proves Schmidt-rank contraction under product operators. Normalizing a nonzero branch does not change matrix rank.

For an adaptive LOCC protocol, a complete classical transcript selects one local Kraus operator at each stage. Multiplying the local operators along that transcript still gives one product $A_t\otimes B_t$. ConsequentlyClassical communication changes which product operator is chosen, not this rank bound. If the overall output is a pure state, every nonzero branch must be proportional to that same vector, so its Schmidt rank also cannot increase. If the outcome is discarded and the output is mixed, pure-state Schmidt rank is not defined; the induced decomposition instead shows that its Schmidt number is at most the original rank. This includes probabilistic filtering as well as deterministic pure-state conversion.