Let $\rho,\sigma$ be a pair of bipartite quantum states such that ${\rm Tr}_1\rho={\rm Tr}_1\sigma$. What does this tell us about $\rho,\sigma$? More precisely, is there a way to write more explicitly the relation between these two states?
In the special case in which they are pure, this is doable in a relatively straightforward manner via Schmidt decomposition: write $$|\psi\rangle = \sum_k \sqrt{p_k}(|u_k\rangle\otimes |v_k\rangle), \qquad |\psi'\rangle = \sum_k \sqrt{p_k'}(|u_k'\rangle\otimes |v_k'\rangle),$$ and then equality of partial traces implies that (up to some reordering of the elements in the sums) $p_k=p_k'$ and $|v_k\rangle=|v_k'\rangle$, while there is no additional constraint on $|u_k'\rangle$. So this provides a relatively nice way to see how the equality of partial traces condition reflects on the structure of the states, and we can write that in general two such states are related by $|\psi\rangle=(U\otimes I)|\psi'\rangle$ for some unitary $U$.
Is there any similar type of "nice" characterisation doable for more general mixed states?