I was trying to understand the concept of entanglement entropy and for that I was studying the density operator formalism in the context of a bipartite quantum system.
In some references, people define the maximally entangled state as the one whose reduced density matrix (for the lower dimensional subsystem) is maximally mixed. I believe I understand the concept of maximally mixed - it just means that the state is descibed by an ensemble with every possible quantum state having the same probability (in some sense it describes the most random ensemble one can have with a certain basis of quantum states). Now what I don't understand in a clear way is why we define a maximally entangled state in such a way - is there any more insightful reason or is it just a definition that is consistent?
Indeed, this question arised when I was thinking about entanglement entropy. Still in the bipartite system, one can show that for a pure state of the system, the Von Neumann entropy of the subsystem has some properties that lead us to define this entropy as a measure of entanglement, yet these same properties are dependent on our notion of maximally entangled that I don't understand (http://www.mpmueller.net/seminar/talk2.pdf see Theorem.6)
Summing up, my questions are:
- When one defines maximal entanglement, do we already have entropy in mind which, indeed, quantifies entanglement? If not, why can one talk about maximal entanglement in some state a priori?
- Is there any way to see that Von Neumann entropy for a subsystem of a pure bipartite state really is quantifying entanglement in a more concrete way? Well, this is my main question, because although I see that this is plausible I don't understand clearly why.