Remark: Why no universal Stein upper bound follows from trace distance alone

Consider pinching-related output states

where $|{+}_k⟩=k^{-1/2}{{\sum }_{r=1}}^k|r⟩$. They can be obtained from a pure recorded preparation by making the sector flag classical at the interface. Their values are

For fixed $0<{\delta }_0<1$, choose $t={\delta }_0/(1-1/k)$ for sufficiently large $k$. The trace distance is exactly ${\delta }_0$, while relative entropy grows without bound. One may take $k$ to be powers of two to realize the example using qubit registers.