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In "SEPARABILITY CRITERION FOR MULTIPARTITE QUANTUM STATES BASED ON THE BLOCH REPRESENTATION OF DENSITY MATRICES" by Hassan and Joag, I found this remarkable thing about entanglement of mixed GHZ states, saying:

We consider [a] $N$-qubit state $$\rho_{noisy}^{N} = \frac{1-p}{2^N}I + p|\psi\rangle\langle\psi|, 0\le p\le1 $$ where $|\psi\rangle$ is a $N$-qubit ... GHZ state.

$\hskip2.0in$enter image description here

Does anyone know how these value would evolve when $N$ grows?

Their entanglement/separability criterion is given as Theorem 1.

If a $N$-partite quantum state of dimension $d_1d_2...d_N$ with Bloch representation $(8)$ is fully separable, then $$ ||\mathcal{T}^{(N)}||_{KF}\le \sqrt{\frac1{2^N}\prod_{k=1}^N d_k(d_k-1)} $$

$\mathcal{T}^{(N)}$ is given as last term in $(8)$: enter image description here

$||\mathcal T||_{KF} = max\{||T_{(n)}^N||_{KF}\}, n=1,...,N;$ is a Ky-Fan norm, which is the sum of the $k$ largest singular values of the matrix unfoldings of $\mathcal T$.

I'm not interested in $|W\rangle$ states...

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  • $\begingroup$ GHZ seem to have a downward trend with larger N... not sure about the W though $\endgroup$ Jul 17 '20 at 2:19
  • $\begingroup$ what entanglement criterion is being used here? $\endgroup$
    – glS
    Jul 17 '20 at 6:23
  • $\begingroup$ @HasanIqbal yes, interesting isn't it? I edited my question... $\endgroup$
    – draks ...
    Jul 17 '20 at 7:10
  • $\begingroup$ @glS it is rather a separability criterion. I added the corresponding parts of the paper... $\endgroup$
    – draks ...
    Jul 17 '20 at 7:11
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I don't know the details of this paper, although there are much better things that you can say about the GHZ case (in general, properties of GHZ states are much easier to analyse than W states). I'll summarise the key result in this context below, but further details are available in my paper, here.

There are some very simple entanglement criteria that one can apply. In particular, pick any bipartition of the system. If that bipartition contains entanglement (which might be found using the partial transpose criterion), then the state is certainly not fully separable, because it is not separable across that bipartition. For the GHZ state, this threshold occurs at $$ p=\frac{1}{2^{N-1}+1}. $$ These values are lower than those stated in the table, and so give a stronger claim. It actually turns out that one can prove this is the threshold for the state becoming fully separable, so it's an exact result.

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  • $\begingroup$ +1 thanks, so you're saying that we are able to determine(perform a measurement) that a mixed quantum state $\rho=(1-p)I + p|GHZ_N\rangle\langle GHZ_N|$ is entangled? $\endgroup$
    – draks ...
    Jul 17 '20 at 7:54
  • $\begingroup$ @draks...yes in the paper there's an explicit construction of an entanglement witness that you could measure. $\endgroup$
    – DaftWullie
    Jul 17 '20 at 8:36
  • $\begingroup$ hmm, but your witness $W_{x,z}$ (if we speak about this, do we?) asks for two choices of $x,z\in\{0,1\}^N$. How to pick the right ones? there are exponentially many... $\endgroup$
    – draks ...
    Jul 17 '20 at 9:34
  • $\begingroup$ @draks... There could be, but they won't be hard to pick in this case because it's such a simple state. I'm a bit too rusty (and focussed on other things) to give you an instant answer. $\endgroup$
    – DaftWullie
    Jul 17 '20 at 9:37
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    $\begingroup$ If the state is diagonal in the graph state basis, then there's a condition to check if the result holds. Changing the diagonal entries in the computational basis probably makes it not diagonal in the graph state basis. $\endgroup$
    – DaftWullie
    Jul 17 '20 at 10:52

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