I would like to know if there are inequalities related to the purity of the partial trace of a bipartite system. The purity $P$ of a density matrix $\rho$ is given by $$P(\rho) = Tr(\rho^2).$$ The most trivial inequality is $1/n \leq P \leq 1$, where $n$ is the dimension of a system.
Given a bipartite system $AB$ and the respective density matrix $\rho$, I would like to know if there are any inequalities or formulas relating $P(Tr_B(\rho))$ and $P(Tr_A(\rho))$ or $P(Tr_A(M \rho M^{\dagger}))$ where $M$ is any arbitrary operator.
Any references and links to the list of relations are very much appreciated!
