Simple proof that entangled pure states are not separable

Question

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!

Answer 1

simple proof that pure, entangled states are not separable

You won't find a proof. It's a matter of definition. We define entanglement by the statement "a state is entangled if it is not separable".

Is there a simple example of a state that is neither pure nor separable

Sure! The states $(1-p)I/4+p|\psi\rangle\langle\psi|$ where $|\psi\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$ are easily proven to be entangled for a range of parameters $p$ (exercise: what range?) using the partial transpose criterion.

a state is non-separable only if any of its purification is entangled

If it is non-separable, it is already entangled. No purification required (but yes, a purification would contain entanglement because an action such as tracing out a qubit cannot increase the entanglement present). However, the other direction is not true - if you have an entangled purification, it does not mean that the state is entangled (see next point).

a state is separable if and only if any of its purification is a product

This is false. Any separable mixed state will have an entangled purification. (If the purification is separable, you have a product state. That means that when you trace out the extra system, you're left with a pure separable state. Hence the purification of mixed separable states is not a separable state, and is therefore entangled.)