Minimum of the Schatten 1-norm

Question

Given two operators or non-zero matrices $A$ and $B$, where $A\neq B$, tr$(A)=1$ and tr$(B)=1$ and tr$(A-B)=0$, what is a lower bound of the Schatten p-norm ($p=1$) $\|A-B\|_1$? Any helpful references?

Answer 1

The lower bound is zero. Consider the matrices $$ A = \begin{pmatrix} 1/2 & 0 \\ 0 & 1/2 \end{pmatrix}$$ and $$ B_\lambda = \begin{pmatrix} \lambda & 0 \\ 0 & 1-\lambda \end{pmatrix}.$$ For $\lambda \ne 1/2$ you have $A \ne B_\lambda$ and $\|A - B_\lambda\|_1=2 \, |1/2 - \lambda|$. For $\lambda \to 1/2$, this converges towards $0$.