Pretty much as the title says. I am interested in the two particle system, each particle having two dimensional quantum states; naturally if there is a generalisation I'd be interested in that too. So, the particular problem I am thinking of is: can one find a $4\times 4$ unitary matrix $U$ that always maps a factorizable state to a maximally entangled one and contrariwise?.
By "maximally entangled" I mean that the von Neumann entropy of one particle alone (which in general seems to be in a mixed state owing to its entanglement with the other) is maximal and equal to one bit (I think this is a standard definition: this is not my field).
I am asking this question as part of trying to find ways to visualise the set of maximally entangled and factorizable states in the bipartite system state space.
PS: I am having trouble deciding which tags to put on my question; I'd appreciate help here too.