I have a conjecture on Schatten 1-norm.
Before presenting the conjecture, let us first specify the notions used here. A matrix $A$ is said to be a density operator if $A$ is positive semidefinite with $\textrm{tr}(A)=1$. The Schatten 1-norm of a matrix $A$ is $\|A\|_1:=\textrm{tr}\sqrt{A^{*}A}$.
My conjecture is: if $\|\rho_1\otimes\sigma_1-\rho_2\otimes\sigma_2\|_1=\|\rho_1-\rho_2\|_1$ and $\rho_1\rho_2\neq0$, i.e., the supports of $\rho_1$ and $\rho_2$ are not orthogonal, then $\sigma_1=\sigma_2$, where $\rho_i$, $\sigma_i$, $i=1,2$ are density operators.
I know this conjecture holds for some special cases, but I do not know how to prove it in general. So can anyone help me?