I am trying to understand more about the notion of separable states. For clarity, I will only use the word entangled for pure states, even if a non-separable state is sometimes called entangled too.
- Where could I find a simple proof that pure, entangled states are not separable? The only I found is a sketch of proof that involves the notion of RHS rank and I do not understand its meanings. Is it linked to the Schmidt decomposition of the state?
- Is there a simple example of a state that is neither pure nor separable?
- Is it true that a state is non-separable only if any of its purification is entangled? (Same for: separable if and only if any of its purification is a product).
Thanks for your help!