I have a situation where I would like to know something about an operator when I know something about its partial trace.
Let $A$ be a trace class operator on $H\otimes H$, nonnegative, of trace one, and symmetric in the sense that $$P_S\,A\,P_S=A\,,$$ where $P_S$ is the projection onto the subspace of symmetric vectors in $H\otimes H$. ($H$ is a possibly infinite-dimensional Hilbert space.)
Let $a$ be the partial trace of $A$. I know that $a$ is almost the orthogonal projection $p_v$ on a unit vector $v$ in $H$: $$Tr|a-p_v|\ll 1\,.$$
I would like to prove that $$Tr|A-p_v\otimes p_v|\ll 1\,.$$ Does anybody have any ideas about how to show this?