Lemma: Pinching domination

Let $k$ be the number of record-$J$ sectors with nonzero weight in $|\Psi ⟩$. Then $1\le k\le 2^{|J|}$ and

Consequently,

Proof. Write $|\Psi ⟩={{\sum }_{z=1}}^k|v_z⟩$ in orthogonal record sectors. For every vector $u$, Cauchy–Schwarz gives

which is the first operator inequality. A positive channel preserves it.

For $k>1$,

for a state $\tau$, so $\delta \le 1-1/k$. For $k=1$, the states coincide. The support inclusion also follows from domination.

Operator monotonicity of the logarithm, with a positive regularization and then a limit on the support of $\varrho$, gives

This proves the relative-entropy bound. The underlying pinching inequality is standard OgawaHayashi2002.