1
$\begingroup$

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!

$\endgroup$
2
  • $\begingroup$ RHS refers to "right-hand side", as in the expression on the right side of the '=' sign in an equation that is being referred to. $\endgroup$
    – forky40
    May 22, 2023 at 22:51
  • $\begingroup$ related: physics.stackexchange.com/a/643655/58382 $\endgroup$
    – glS
    May 22, 2023 at 23:32

1 Answer 1

2
$\begingroup$

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.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.